Skip to main content
SpringerLink
Log in

Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Computation of incompressible flows in arterial fluid mechanics, especially because it involves fluid–structure interaction, poses significant numerical challenges. Iterative solution of the fluid mechanics part of the equation systems involved is one of those challenges, and we address that in this paper, with the added complication of having boundary layer mesh refinement with thin layers of elements near the arterial wall. As test case, we use matrix data from stabilized finite element computation of a bifurcating middle cerebral artery segment with aneurysm. It is well known that solving linear systems that arise in incompressible flow computations consume most of the time required by such simulations. For solving these large sparse nonsymmetric systems, we present effective preconditioning techniques appropriate for different stages of the computation over a cardiac cycle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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 

  2. 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: 1885–1895

    Article  MATH  MathSciNet  Google Scholar 

  3. 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: 482–490

    Article  MATH  Google Scholar 

  4. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322

    Article  MATH  MathSciNet  Google Scholar 

  5. Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2007) Influence of wall elasticity in patient-specific hemodynamic simulations. Comput Fluids 36: 160–168

    Article  MATH  Google Scholar 

  6. 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: 901–922

    Article  MATH  MathSciNet  Google Scholar 

  7. 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: 995–1009

    Article  MATH  MathSciNet  Google Scholar 

  8. 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: 601–629

    Article  MATH  MathSciNet  Google Scholar 

  9. Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43: 3–37

    Article  MATH  MathSciNet  Google Scholar 

  10. 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: 151–159

    Article  MATH  Google Scholar 

  11. Tezduyar TE, Schwaab M, Sathe S (2008) Sequentially-coupled arterial fluid–structure interaction (SCAFSI) technique. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2008.05.024

  12. Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2008) Fluid–structure interaction modeling of blood flow and cerebral aneurysm: significance of artery and aneurysm shapes. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2008.08.020

  13. 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. doi:10.1016/j.cma.2009.04.015

  14. Takizawa K, Christopher J, Tezduyar TE, Sathe S (2009) Space–time finite element computation of arterial fluid–structure interactions with patient-specific data. Commun Numer Methods Eng. doi:10.1002/cnm.1241

  15. 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, Kassel

    Google Scholar 

  16. Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2009) Influence of wall thickness on fluid–structure interaction computations of cerebral aneurysms. Commun Numer Methods Eng. doi:10.1002/cnm.1289

  17. Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26: 27–36

    Article  Google Scholar 

  18. 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: 157–177

    Article  MATH  Google Scholar 

  19. Mittal S, Tezduyar TE (1994) Massively parallel finite element computation of incompressible flows involving fluid–body interactions. Comput Methods Appl Mech Eng 112: 253–282

    Article  MATH  MathSciNet  Google Scholar 

  20. Mittal S, Tezduyar TE (1995) Parallel finite element simulation of 3D incompressible flows—fluid–structure interactions. Int J Numer Methods Fluids 21: 933–953

    Article  MATH  Google Scholar 

  21. Johnson AA, Tezduyar TE (1997) Parallel computation of incompressible flows with complex geometries. Int J Numer Methods Fluids 24: 1321–1340

    Article  MATH  Google Scholar 

  22. Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23: 130–143

    Article  MATH  Google Scholar 

  23. Kalro V, Tezduyar TE (2000) A parallel 3D computational method for fluid–structure interactions in parachute systems. Comput Methods Appl Mech Eng 190: 321–332

    Article  MATH  Google Scholar 

  24. 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: 373–386

    Article  MATH  Google Scholar 

  25. Tezduyar T, Osawa Y (2001) Fluid–structure interactions of a parachute crossing the far wake of an aircraft. Comput Methods Appl Mech Eng 191: 717–726

    Article  MATH  Google Scholar 

  26. Ohayon R (2001) Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems. Comput Methods Appl Mech Eng 190: 3009–3019

    Article  MATH  Google Scholar 

  27. Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid–structure interactions with large displacements. J Appl Mech 70: 58–63

    Article  MATH  Google Scholar 

  28. 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, Mexico

  29. Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193: 2019–2032

    Article  MATH  Google Scholar 

  30. van Brummelen EH, de Borst R (2005) On the nonnormality of subiteration for a fluid–structure interaction problem. SIAM J Sci Comput 27: 599–621

    Article  MATH  MathSciNet  Google Scholar 

  31. Michler C, van Brummelen EH, de Borst R (2005) An interface Newton–Krylov solver for fluid–structure interaction. Int J Numer Methods Fluids 47: 1189–1195

    Article  MATH  Google Scholar 

  32. 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: 2002–2027

    Article  MATH  MathSciNet  Google Scholar 

  33. 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: 5743–5753

    Article  MATH  MathSciNet  Google Scholar 

  34. Tezduyar TE, Sathe S, Stein K, Aureli L (2006) Modeling of fluid–structure interactions with the space–time techniques. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction. Lecture notes in computational science and engineering, vol 53. Springer, Berlin, pp 50–81

  35. Dettmer W, Peric D (2006) A computational framework for fluid–structure interaction: finite element formulation and applications. Comput Methods Appl Mech Eng 195: 5754–5779

    Article  MATH  Google Scholar 

  36. 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: 403–416

    Article  MATH  Google Scholar 

  37. Kuttler U, Forster C, Wall WA (2006) A solution for the incompressibility dilemma in partitioned fluid–structure interaction with pure Dirichlet fluid domains. Comput Mech 38: 417–429

    Article  Google Scholar 

  38. Bletzinger K-U, Wuchner R, Kupzok A (2006) Algorithmic treatment of shells and free form-membranes in FSI. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction. Lecture notes in computational science and engineering, vol 53. Springer, Berlin, pp 336–355

  39. Masud A, Bhanabhagvanwala M, Khurram RA (2007) An adaptive mesh rezoning scheme for moving boundary flows and fluid–structure interaction. Comput Fluids 36: 77–91

    Article  MATH  MathSciNet  Google Scholar 

  40. Sawada T, Hisada T (2007) Fuid–structure interaction analysis of the two dimensional flag-in-wind problem by an interface tracking ALE finite element method. Comput Fluids 36: 136–146

    Article  MATH  Google Scholar 

  41. Wall WA, Genkinger S, Ramm E (2007) A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Comput Fluids 36: 169–183

    Article  MATH  Google Scholar 

  42. Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: solution techniques. Int J Numer Methods Fluids 54: 855–900

    Article  MATH  MathSciNet  Google Scholar 

  43. Kuttler U, Wall WA (2008) Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput Mech 43: 61–72

    Article  Google Scholar 

  44. Dettmer WG, Peric D (2008) On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction. Comput Mech 43: 81–90

    Article  MATH  Google Scholar 

  45. 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: 39–49

    Article  MATH  Google Scholar 

  46. Tezduyar TE, Sathe S, Pausewang J, Schwaab M, Christopher J, Crabtree J (2008) Fluid–structure interaction modeling of ringsail parachutes. Comput Mech 43: 133–142

    Article  MATH  Google Scholar 

  47. Sathe S, Tezduyar TE (2008) Modeling of fluid–structure interactions with the space–time finite elements: contact problems. Comput Mech 43: 51–60

    Article  MATH  MathSciNet  Google Scholar 

  48. Manguoglu M, Sameh AH, Tezduyar TE, Sathe S (2008) A nested iterative scheme for computation of incompressible flows in long domains. Comput Mech 43: 73–80

    Article  MATH  MathSciNet  Google Scholar 

  49. Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28: 1–44

    Article  MATH  MathSciNet  Google Scholar 

  50. 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: 339–351

    Article  MATH  MathSciNet  Google Scholar 

  51. 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: 353–371

    Article  MATH  MathSciNet  Google Scholar 

  52. Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43: 555–575

    Article  MATH  MathSciNet  Google Scholar 

  53. Takizawa K, Moorman C, Wright S, Christopher J, Tezduyar TE (2009) Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions. Comput Mech (submitted)

  54. Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th national conference. ACM Press, New York, pp 157–172

  55. Baggag A, Sameh A (2004) A nested iterative scheme for indefinite linear systems in particulate flows. Comput Methods Appl Mech Eng 193: 1923–1957

    Article  MATH  MathSciNet  Google Scholar 

  56. Sameh A, Manguoglu M, Sathe S, Tezduyar TE (2007) A nested iterative scheme for nonsymmetric linear systems. In: Onate E, Papadrakakis M, Schrefler B (eds) Coupled problems 2007. CIMNE, Barcelona

    Google Scholar 

  57. Sameh A, Manguoglu M, Sathe S, Pausewang J, Tezduyar TE (2007) Iterative techniques with banded preconditioners for fluid mechanics computations over long domains. In: Onate E, Garcia J, Bergan P, Kvamsdal T (eds) Marine 2007. CIMNE, Barcelona

    Google Scholar 

  58. Sameh AH, Manguoglu M, Sathe S, Pausewang J, Tezduyar TE (2007) Iterative schemes for time accurate solution of flow in long narrow domains. In: Proceedings of the third Asian-Pacific congress on computational mechanics (CD-ROM), Kyoto, Japan

  59. 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: 021204

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Purdue University, 305 N. University Street, West Lafayette, IN, 47907-2107, USA

    Murat Manguoglu & Ahmed H. Sameh

  2. Mechanical Engineering, Rice University, MS 321, 6100 Main Street, Houston, TX, 77005, USA

    Kenji Takizawa & Tayfun E. Tezduyar

Authors
  1. Murat Manguoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenji Takizawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed H. Sameh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tayfun E. Tezduyar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed H. Sameh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Manguoglu, M., Takizawa, K., Sameh, A.H. et al. Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement. Comput Mech 46, 83–89 (2010). https://doi.org/10.1007/s00466-009-0426-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-009-0426-z

Keywords

Search

Navigation