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Computational Mechanics

, Volume 48, Issue 3, pp 247–267 | Cite as

Multiscale space–time fluid–structure interaction techniques

  • Kenji Takizawa
  • Tayfun E. Tezduyar
Original Paper

Abstract

We present the multiscale space–time techniques we have developed for fluid–structure interaction (FSI) computations. Some of these techniques are multiscale in the way the time integration is performed (i.e. temporally multiscale), some are multiscale in the way the spatial discretization is done (i.e. spatially multiscale), and some are in the context of the sequentially-coupled FSI (SCFSI) techniques developed by the Team for Advanced Flow Simulation and Modeling \({({\rm T} \bigstar {\rm AFSM})}\). In the multiscale SCFSI technique, the FSI computational effort is reduced at the stage we do not need it and the accuracy of the fluid mechanics (or structural mechanics) computation is increased at the stage we need accurate, detailed flow (or structure) computation. As ways of increasing the computational accuracy when or where needed, and beyond just increasing the mesh refinement or decreasing the time-step size, we propose switching to more accurate versions of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation, using more polynomial power for the basis functions of the spatial discretization or time integration, and using an advanced turbulence model. Specifically, for more polynomial power in time integration, we propose to use NURBS, and as an advanced turbulence model to be used with the DSD/SST formulation, we introduce a space–time version of the residual-based variational multiscale method. We present a number of test computations showing the performance of the multiscale space–time techniques we are proposing. We also present a stability and accuracy analysis for the higher-accuracy versions of the DSD/SST formulation.

Keywords

Fluid–structure interaction Space–time formulations Multiscale techniques Sequential coupling techniques NURBS Space–time variational multiscale method 

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References

  1. 1.
    Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29: 329–349MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26: 27–36CrossRefGoogle Scholar
  3. 3.
    Tezduyar TE, Aliabadi SK, Behr M, Mittal S (1994) Massively parallel finite element simulation of compressible and incompressible flows. Comput Methods Appl Mech Eng 119: 157–177zbMATHCrossRefGoogle Scholar
  4. 4.
    Mittal S, Tezduyar TE (1994) Massively parallel finite element computation of incompressible flows involving fluid–body interactions. Comput Methods Appl Mech Eng 112: 253–282MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Mittal S, Tezduyar TE (1995) Parallel finite element simulation of 3D incompressible flows—fluid–structure interactions. Int J Numer Methods Fluids 21: 933–953zbMATHCrossRefGoogle Scholar
  6. 6.
    Johnson AA, Tezduyar TE (1997) Parallel computation of incompressible flows with complex geometries. Int J Numer Methods Fluids 24: 1321–1340zbMATHCrossRefGoogle Scholar
  7. 7.
    Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23: 130–143zbMATHCrossRefGoogle Scholar
  8. 8.
    Kalro V, Tezduyar TE (2000) A parallel 3D computational method for fluid–structure interactions in parachute systems. Comput Methods Appl Mech Eng 190: 321–332zbMATHCrossRefGoogle Scholar
  9. 9.
    Stein K, Benney R, Kalro V, Tezduyar TE, Leonard J, Accorsi M (2000) Parachute fluid–structure interactions: 3-D computation. Comput Methods Appl Mech Eng 190: 373–386zbMATHCrossRefGoogle Scholar
  10. 10.
    Tezduyar T, Osawa Y (2001) Fluid–structure interactions of a parachute crossing the far wake of an aircraft. Comput Methods Appl Mech Eng 191: 717–726zbMATHCrossRefGoogle Scholar
  11. 11.
    Ohayon R (2001) Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems. Comput Methods Appl Mech Eng 190: 3009–3019zbMATHCrossRefGoogle Scholar
  12. 12.
    Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid–structure interactions with large displacements. J Appl Mech 70: 58–63zbMATHCrossRefGoogle Scholar
  13. 13.
    Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193: 2019–2032zbMATHCrossRefGoogle Scholar
  14. 14.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2004) Influence of wall elasticity on image-based blood flow simulation. Jpn Soc Mech Eng J Ser A 70: 1224–1231 (in Japanese)Google Scholar
  15. 15.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2004) Space–time techniques for finite element computation of flows with moving boundaries and interfaces. In: Gallegos S, Herrera I, Botello S, Zarate F, Ayala G (eds) Proceedings of the III international congress on numerical methods in engineering and applied science, CD-ROM, Monterrey, MexicoGoogle Scholar
  16. 16.
    van Brummelen EH, de Borst R (2005) On the nonnormality of subiteration for a fluid-structure interaction problem. SIAM J Sci Comput 27: 599–621MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space–time finite element techniques for computation of fluid–structure interactions. Comput Methods Appl Mech Eng 195: 2002–2027MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Tezduyar TE, Sathe S, Stein K (2006) Solution techniques for the fully-discretized equations in computation of fluid–structure interactions with the space–time formulations. Comput Methods Appl Mech Eng 195: 5743–5753MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Computer modeling of cardiovascular fluid–structure interactions with the Deforming-Spatial-Domain/Stabilized Space–Time formulation. Comput Methods Appl Mech Eng 195: 1885–1895MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Fluid–structure interaction modeling of aneurysmal conditions with high and normal blood pressures. Comput Mech 38: 482–490zbMATHCrossRefGoogle Scholar
  21. 21.
    Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Khurram RA, Masud A (2006) A multiscale/stabilized formulation of the incompressible Navier–Stokes equations for moving boundary flows and fluid–structure interaction. Comput Mech 38: 403–416zbMATHCrossRefGoogle Scholar
  23. 23.
    Tezduyar TE, Sathe S, Cragin T, Nanna B, Conklin BS, Pausewang J, Schwaab M (2007) Modeling of fluid–structure interactions with the space–time finite elements: arterial fluid mechanics. Int J Numer Methods Fluids 54: 901–922MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2007) Influence of wall elasticity in patient-specific hemodynamic simulations. Comput Fluids 36: 160–168zbMATHCrossRefGoogle Scholar
  25. 25.
    Sawada T, Hisada T (2007) Fluid–structure interaction analysis of the two dimensional flag-in-wind problem by an interface tracking ALE finite element method. Comput Fluids 36: 136–146zbMATHCrossRefGoogle Scholar
  26. 26.
    Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: solution techniques. Int J Numer Methods Fluids 54: 855–900MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2007) Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm—dependence of the effect on the aneurysm shape. Int J Numer Methods Fluids 54: 995–1009MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Manguoglu M, Sameh AH, Tezduyar TE, Sathe S (2008) A nested iterative scheme for computation of incompressible flows in long domains. Comput Mech 43: 73–80MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Tezduyar TE, Sathe S, Pausewang J, Schwaab M, Christopher J, Crabtree J (2008) Interface projection techniques for fluid– structure interaction modeling with moving-mesh methods. Comput Mech 43: 39–49zbMATHCrossRefGoogle Scholar
  30. 30.
    Tezduyar TE, Sathe S, Schwaab M, Conklin BS (2008) Arterial fluid mechanics modeling with the stabilized space–time fluid–structure interaction technique. Int J Numer Methods Fluids 57: 601–629MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2008) Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling. Comput Mech 43: 151–159zbMATHCrossRefGoogle Scholar
  32. 32.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43: 3–37MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Isaksen JG, Bazilevs Y, Kvamsdal T, Zhang Y, Kaspersen JH, Waterloo K, Romner B, Ingebrigtsen T (2008) Determination of wall tension in cerebral artery aneurysms by numerical simulation. Stroke 39: 3172–3178CrossRefGoogle Scholar
  34. 34.
    Kuttler U, Wall WA (2008) Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput Mech 43: 61–72CrossRefGoogle Scholar
  35. 35.
    Dettmer WG, Peric D (2008) On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction. Comput Mech 43: 81–90zbMATHCrossRefGoogle Scholar
  36. 36.
    Tezduyar TE, Schwaab M, Sathe S (2009) Sequentially-Coupled Arterial Fluid–Structure Interaction (SCAFSI) technique. Comput Methods Appl Mech Eng 198: 3524–3533MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2009) Fluid–structure interaction modeling of blood flow and cerebral aneurysm: significance of artery and aneurysm shapes. Comput Methods Appl Mech Eng 198: 3613–3621MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Manguoglu M, Sameh AH, Saied F, Tezduyar TE, Sathe S (2009) Preconditioning techniques for nonsymmetric linear systems in computation of incompressible flows. J Appl Mech 76: 021204CrossRefGoogle Scholar
  39. 39.
    Bazilevs Y, Gohean JR, Hughes TJR, Moser RD, Zhang Y (2009) Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Comput Methods Appl Mech Eng 198: 3534–3550MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Bazilevs Y, Hsu M-C, Benson D, Sankaran S, Marsden A (2009) Computational fluid–structure interaction: methods and application to a total cavopulmonary connection. Comput Mech 45: 77–89MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Takizawa K, Christopher J, Tezduyar TE, Sathe S (2010) Space–time finite element computation of arterial fluid–structure interactions with patient-specific data. Int J Numer Methods Biomed Eng 26: 101–116zbMATHCrossRefGoogle Scholar
  42. 42.
    Tezduyar TE, Takizawa K, Christopher J (2009) Multiscale Sequentially-Coupled Arterial Fluid–Structure Interaction (SCAFSI) technique. In: Hartmann S, Meister A, Schaefer M, Turek S (eds) International workshop on fluid–structure interaction—theory, numerics and applications, Kassel University Press, pp 231–252Google Scholar
  43. 43.
    Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Multiscale sequentially-coupled arterial FSI technique. Comput Mech 46: 17–29MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Takizawa K, Moorman C, Wright S, Christopher J, Tezduyar TE (2010) Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions. Comput Mech 46: 31–41MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2010) Influence of wall thickness on fluid–structure interaction computations of cerebral aneurysms. Int J Numer Methods Biomed Eng 26: 336–347MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Manguoglu M, Takizawa K, Sameh AH, Tezduyar TE (2010) Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement. Comput Mech 46: 83–89zbMATHCrossRefGoogle Scholar
  47. 47.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2010) Role of 0D peripheral vasculature model in fluid–structure interaction modeling of aneurysms. Comput Mech 46: 43–52zbMATHCrossRefGoogle Scholar
  48. 48.
    Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Liang X, Kvamsdal T, Brekken R, Isaksen J (2010) A fully-coupled fluid–structure interaction simulation of cerebral aneurysms. Comput Mech 46: 3–16MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space–time finite element computation of complex fluid–structure interactions. Int J Numer Methods Fluids 64: 1201–1218zbMATHCrossRefGoogle Scholar
  50. 50.
    Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Kvamsdal T, Hentschel S, Isaksen J (2010) Computational fluid–structure interaction: methods and application to cerebral aneurysms. Biomech Model Mechanobiol 9: 481–498CrossRefGoogle Scholar
  51. 51.
    Bazilevs Y, Hsu M-C, Akkerman I, Wright S, Takizawa K, Henicke B, Spielman T, Tezduyar TE (2011) 3D simulation of wind turbine rotors at full scale. Part I. Geometry modeling and aerodynamics. Int J Numer Methods Fluids 65: 207–235zbMATHCrossRefGoogle Scholar
  52. 52.
    Bazilevs Y, Hsu M-C, Kiendl J, Wüchner R, Bletzinger K-U (2011) 3D simulation of wind turbine rotors at full scale. Part II. Fluid–structure interaction modeling with composite blades. Int J Numer Methods Fluids 65: 236–253zbMATHCrossRefGoogle Scholar
  53. 53.
    Takizawa K, Moorman C, Wright S, Spielman T, Tezduyar TE (2011) Fluid–structure interaction modeling and performance analysis of the Orion spacecraft parachutes. Int J Numer Methods Fluids 65: 271–285zbMATHCrossRefGoogle Scholar
  54. 54.
    Takizawa K, Wright S, Moorman C, Tezduyar TE (2011) Fluid–structure interaction modeling of parachute clusters. Int J Numer Methods Fluids 65: 286–307zbMATHCrossRefGoogle Scholar
  55. 55.
    Manguoglu M, Takizawa K, Sameh AH, Tezduyar TE (2011) Nested and parallel sparse algorithms for arterial fluid mechanics computations with boundary layer mesh refinement. Int J Numer Methods Fluids 65: 135–149MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Tezduyar TE, Takizawa K, Brummer T, Chen PR (2011) Space–time fluid–structure interaction modeling of patient-specific cerebral aneurysms. Int J Numer Methods Biomed Eng (in press)Google Scholar
  57. 57.
    Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28: 1–44MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space–time procedure. I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94: 339–351MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Tezduyar TE, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space–time procedure. II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94: 353–371MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43: 555–575MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov- Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32: 199–259MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal- order-interpolation velocity–pressure elements. Comput Methods Appl Mech Eng 95: 221–242zbMATHCrossRefGoogle Scholar
  63. 63.
    Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Hughes TJR, Hulbert GM (1988) Space–time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Eng 66: 339–363MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Tezduyar TE, Sathe S, Pausewang J, Schwaab M, Christopher J, Crabtree J (2008) Fluid–structure interaction modeling of ringsail parachutes. Comput Mech 43: 133–142zbMATHCrossRefGoogle Scholar
  66. 66.
    Takizawa K, Moorman C, Wright S, Purdue J, McPhail T, Chen PR, Warren J, Tezduyar TE (2011) Patient-specific arterial fluid–structure interaction modeling of cerebral aneurysms. Int J Numer Methods Fluids 65: 308–323zbMATHCrossRefGoogle Scholar
  67. 67.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194: 4135–4195MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Bazilevs Y, Hughes TJR (2008) NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput Mech 43: 143–150MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles, and the origins of stabilized methods. Comput Methods Appl Mech Eng 127: 387–401zbMATHCrossRefGoogle Scholar
  70. 70.
    Hughes TJR, Oberai AA, Mazzei L (2001) Large eddy simulation of turbulent channel flows by the variational multiscale method. Phys Fluids 13: 1784–1799CrossRefGoogle Scholar
  71. 71.
    Hughes TJR, Sangalli G (2007) Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J Numer Anal 45: 539–557MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Bazilevs Y, Calo VM, Cottrel JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197: 173–201zbMATHCrossRefGoogle Scholar
  73. 73.
    Bazilevs Y, Akkerman I (2010) Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residual–based variational multiscale method. J Comput Phys 229: 3402–3414MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Tezduyar TE, Takizawa K, Christopher J (2009) Sequentially- coupled FSI technique. In: Kvamsdal T, Pettersen B, Bergan P, Onate E, Garcia J (eds) Marine 2009, CIMNE, Barcelona, SpainGoogle Scholar
  75. 75.
    Tezduyar TE, Takizawa K, Christopher J, Moorman C, Wright S (2009) Interface projection techniques for complex FSI problems. In: Kvamsdal T, Pettersen B, Bergan P, Onate E, Garcia J (eds) Marine 2009, CIMNE, Barcelona, Spain, 2009Google Scholar
  76. 76.
    Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190: 411–430zbMATHCrossRefGoogle Scholar
  77. 77.
    Akin JE, Tezduyar T, Ungor M, Mittal S (2003) Stabilization parameters and Smagorinsky turbulence model. J Appl Mech 70: 2–9zbMATHCrossRefGoogle Scholar
  78. 78.
    Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces. In: Stein E, Borst RD, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3 Fluids, chapt 17. Wiley, New YorkGoogle Scholar
  79. 79.
    Akin JE, Tezduyar TE (2004) Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements. Comput Methods Appl Mech Eng 193: 1909–1922zbMATHCrossRefGoogle Scholar
  80. 80.
    Tezduyar TE (2007) Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids 36: 191–206MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Rispoli F, Corsini A, Tezduyar TE (2007) Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Comput Fluids 36: 121–126zbMATHCrossRefGoogle Scholar
  82. 82.
    Catabriga L, Coutinho ALGA, Tezduyar TE (2005) Compressible flow SUPG parameters computed from element matrices. Commun Numer Methods Eng 21: 465–476MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Catabriga L, Coutinho ALGA, Tezduyar TE (2006) Compressible flow SUPG parameters computed from degree-of-freedom submatrices. Comput Mech 38: 334–343zbMATHCrossRefGoogle Scholar
  84. 84.
    Hsu M-C, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2010) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Methods Appl Mech Eng 199: 828–840MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Corsini A, Rispoli F, Tezduyar TE (2011) Stabilized finite element computation of NOx emission in aero-engine combustors. Int J Numer Methods Fluids 65: 254–270MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Tezduyar TE, Park YJ (1986) Discontinuity capturing finite element formulations for nonlinear convection–diffusion-reaction equations. Comput Methods Appl Mech Eng 59: 307–325zbMATHCrossRefGoogle Scholar
  87. 87.
    Shakib F, Hughes TJR, Johan Z (1991) A new finite element formulation for computational fluid dynamics. X. The compressible euler and Navier–Stokes equations. Comput Methods Appl Mech Eng 89: 141–219MathSciNetCrossRefGoogle Scholar
  88. 88.
    Hughes TJR, Oberai AA (2003) Calculation of shear stress in Fourier–Galerkin formulations of turbulent channel flows: projection, the Dirichlet filter and conservation. J Comput Phys 188: 281–295MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Shakib F, Hughes TJR (1991) A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space–time and Galerkin/least-squares algorithms. Comput Methods Appl Mech Eng 87: 35–58MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45: 217–284MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Hughes TJR (1987) The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJzbMATHGoogle Scholar
  92. 92.
    Hauke G, Doweidar MH (2005) Fourier analysis of semi-discrete and space–time stabilized methods for the advective-diffusive-reactive equation. I. SUPG. Comput Methods Appl Mech Eng 194: 45–81MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Karypis G, Kumar V (1999) Parallel multilevel k-way partitioning scheme for irregular graphs. SIAM J Sci Comput 41: 278–300MathSciNetzbMATHGoogle Scholar
  94. 94.
    Saad Y, Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Zalesak ST (1979) Fully multidimensional flux-corrected transport algorithms for fluids. J Comput Phys 31: 335–362MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Timmer WA (2009) An overview of NACA 6-digit airfoil series characteristics with reference to airfoils for large wind turbine blades. In: Proceedings of AIAA 47th aerospace sciences meeting, AIAA Paper 2009-268, Orlando, FLGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Modern Mechanical Engineering, Waseda Institute for Advanced StudyWaseda UniversityShinjuku-ku, TokyoJapan
  2. 2.Department of Mechanical EngineeringRice UniversityHoustonUSA

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