Skip to main content
Log in

Fixed-point fluid–structure interaction solvers with dynamic relaxation

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

Abstract

A fixed-point fluid–structure interaction (FSI) solver with dynamic relaxation is revisited. New developments and insights gained in recent years motivated us to present an FSI solver with simplicity and robustness in a wide range of applications. Particular emphasis is placed on the calculation of the relaxation parameter by both Aitken’s \({\Delta^{2}}\) method and the method of steepest descent. These methods have shown to be crucial ingredients for efficient FSI simulations.

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

Access this article

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. Badia S, Codina R (2007) On some fluid–structure iterative algorithms using pressure segregation methods. application to aeroelasticity. Int J Numer Methods Eng 72(1): 46–71

    Article  MathSciNet  Google Scholar 

  2. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38(4–5): 310–322

    Article  MathSciNet  Google Scholar 

  3. Causin P, Gerbeau J-F, Nobile F (2005) Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput Methods Appl Mech Eng 194: 4506–4527

    Article  MATH  MathSciNet  Google Scholar 

  4. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\({\alpha}\) method. J Appl Math 60: 371–375

    MATH  MathSciNet  Google Scholar 

  5. Deparis S (2004) Numerical analysis of axisymmetric flows and methods for fluid–structure interaction arising in blood flow simulation. Dissertation, EPFL

  6. Deparis S, Discacciati M, Fourestey G, Quarteroni A (2006) Fluid–structure algorithms based on Steklov-Poincaré operators. Comput Methods Appl Mech Eng 195: 5797–5812

    Article  MATH  MathSciNet  Google Scholar 

  7. Dettmer WG, 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 

  8. Farhat C (2004) CFD-based nonlinear computational aeroelasticity. In: Stein E, De Borst R, Hughes TJR(eds) Encyclopedia of Computational mechanics, vol 3, chap. 13. Wiley, NY

    Google Scholar 

  9. Farhat C, Geuzaine P (2004) Design and analysis of robust ale time-integrators for the solution of unsteady flow problems on moving grids. Comput Methods Appl Mech Eng 193: 4073–4095

    Article  MATH  MathSciNet  Google Scholar 

  10. Fernández MÁ, Moubachir M (2005) A Newton method using exact jacobians for solving fluid–structure coupling. Comput Struct 83(2–3): 127–142

    Article  Google Scholar 

  11. Förster Ch (2007) Robust methods for fluid–structure interaction with stabilised finite elements. Dissertation, Institut für Baustatik und Baudynamik Universität Stuttgart

  12. Förster Ch, Wall WA, Ramm E (2005) On the geometric conservation law in transient flow calculations on deforming domains. Int J Numer Methods Fluids 50: 1369–1379

    Article  Google Scholar 

  13. Förster Ch, Wall WA, Ramm E (2007) Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput Methods Appl Mech Eng 196: 1278–1293

    Article  Google Scholar 

  14. Förster Ch, Wall WA, Ramm E (2008) Stabilized finite element formulation for incompressible flow on distorted meshes. Int J Numer Methods Fluids (in press)

  15. Gerbeau J-F, Vidrascu M (2003) A quasi-Newton algorithm based on a reduced model for fluid–structure interaction problems in blood flows. Math Model Numer Anal. Math Model Numer Anal 37(4): 631–647

    Article  MATH  MathSciNet  Google Scholar 

  16. Gerbeau J-F, Vidrascu M, Frey P (2005) Fluid–structure interaction in blood flows on geometries coming from medical imaging. Comput Struct 83: 155–165

    Article  Google Scholar 

  17. Golub GH, Van Loan CF (1996) Matrix computations. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  18. Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid–structure interaction problems. Comput Methods Appl Mech Eng 193: 1–23

    Article  MATH  MathSciNet  Google Scholar 

  19. Hübner B, Walhorn E, Dinkler D (2004) A monolithic approach to fluid–structure interaction using space-time finite elements. Comput Methods Appl Mech Eng 193: 2087–2104

    Article  MATH  Google Scholar 

  20. Irons B, Tuck RC (1969) A version of the Aitken accelerator for computer implementation. Int J Numer Methods Eng 1: 275–277

    Article  MATH  Google Scholar 

  21. 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 

  22. Kelley CT (1995) Iterative Methods for linear and nonlinear equations frontiers in applied mathematics. SIAM

  23. Knoll DA, Keyes DE (2004) Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J Comput Phys 193: 357–397

    Article  MATH  MathSciNet  Google Scholar 

  24. Küttler U, Förster Ch, 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 

  25. Matthies HG, Steindorf J (2003) Partitioned strong coupling algorithms for fluid–structure interaction. Comput Struct 81: 805–812

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  27. Mok DP, Wall WA (2001) Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures. In: Wall WA, Bletzinger K-U, Schweitzerhof K (eds) Trends in computational structural mechanics

  28. Park KC, Felippa CA, Ohayon R (2001) Partitioned formulation of internal fluid–structure interaction problems by localized Lagrange multipliers. Comput Methods Appl Mech Eng 190(24–25): 2989–3007

    Article  MATH  Google Scholar 

  29. Quaini A, Quarteroni A (2007) A semi-implicit approach for fluid–structure interaction based on an algebraic fractional step method. Math Models Methods Appl Sci 17(6): 957–983

    Article  MATH  MathSciNet  Google Scholar 

  30. Le Tallec P, Mouro J (2001) Fluid structure interaction with large structural displacements. Comput Methods Appl Mech Eng 190(24–25): 3039–3067

    Article  MATH  Google Scholar 

  31. Tezduyar TE (2007) Finite elements in fluids: Special methods and enhanced solution techniques. Comput Fluids 36: 207–223

    Article  MathSciNet  Google Scholar 

  32. Tezduyar TE (2007) Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids 36: 191–206

    Article  MathSciNet  Google Scholar 

  33. Tezduyar TE, Sathe S (2007) Modelling of fluid–structure interactions with the space-time finite elements: solution techniques. Int J Numer Meth Fluids 54(6–8): 855–900

    Article  MATH  MathSciNet  Google Scholar 

  34. Tezduyar TE, Sathe S, Cragin T, Nanna B, Conklin BS, Pausewang J, Schwaab M (2007) Modelling of fluid–structure interactions with the space-time finite elements: arterial fluid mechanics. Int J Numer Meth Fluids 54(6–8): 901–922

    Article  MATH  MathSciNet  Google Scholar 

  35. 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 

  36. 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 

  37. Vierendeels J (2005) Implicit coupling of partitioned fluid–structure interaction solvers using a reduced order model. AIAA Fluid Dyn Conf Exhib 35: 1–12

    Google Scholar 

  38. Wall WA (1999) Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen. Dissertation, Institut für Baustatik, Universität Stuttgart

  39. Wall WA, Gerstenberger A, Gamnitzer P, Förster Ch, Ramm F (2006) Large deformation fluid–structure interaction—advances in ALE methods and new fixed grid approaches. In: Bungartz H-J, Schäfer M(eds) Fluid–structure interaction: modelling, simulation, optimisation, LNCSE. Springer, Heidelberg

    Google Scholar 

  40. Wall WA, Mok DP, Ramm E (1999) Partitioned analysis approach of the transient coupled response of viscous fluids and flexible structures. In: Wunderlich W (Ed.), Solids, structures and coupled problems in engineering, proceedings of the European conference on computational mechanics ECCM ’99, Munich

  41. Wüchner R, Kupzok A, Bletzinger K-U (2007) A framework for stabilized partitioned analysis of thin membrane-wind interaction. Int J Numer Methods Fluids 54(6–8)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wolfgang A. Wall.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Küttler, U., Wall, W.A. Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput Mech 43, 61–72 (2008). https://doi.org/10.1007/s00466-008-0255-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0255-5

Keywords

Navigation