Skip to main content
SpringerLink
Log in

On Time Discretizations of Fluid-Structure Interactions

  • Conference paper
Multiple Shooting and Time Domain Decomposition Methods

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 9))

Abstract

In this contribution, time discretizations of fluid-structure interactions are considered. We explore two specific complexities: first, the stiffness of the coupled system including different scales of the Navier-Stokes equations of parabolic type and the structure equation of hyperbolic type and second, the problem of moving domains that is inherent to fluid-structure interactions.

Typical moving mesh approaches, such as the arbitrary Lagrangian-Eulerian framework, give rise to nonlinearities and time-derivatives with respect to the mesh-deformation. We derive different time-stepping techniques of Crank-Nicolson type and analyse their stability and approximation properties. Further, we closely look at the dominant time-scales that must be resolved to capture the global dynamics. Moreover, our discussion is supplemented with an analysis of the temporal discretization of Eulerian fixed-mesh approaches for fluid-structure interactions, where the interface between fluid and solid will change from time-step to time-step. Finally, a formulation of parallel multiple shooting methods for fluid-structure interaction is presented.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)

    Article  MATH  Google Scholar 

  2. Babuška, I., Banarjee, U., Osborn, J.E.: Generalized finite element methods: main ideas, results, and perspective. Int. J. Comput. Methods 1, 67–103 (2004)

    Article  MATH  Google Scholar 

  3. Bangerth, W., Geiger, M., Rannacher, R.: Adaptive Galerkin finite element methods for the wave equation. Comput. Methods Appl. Math. 10, 3–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Börgers, C.: A triangulation algorithm for fast elliptic solvers based on domain imbedding. SIAM J. Numer. Anal. 27, 1187–1196 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Braack, M., Richter, T.: Stabilized finite elements for 3-d reactive flows. Int. J. Numer. Math. Fluids 51, 981–999 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6, 109–138 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bristeau, M.O., Glowinski, R., Periaux, J.: Numerical methods for the Navier-Stokes equations. Comput. Phys. Rep. 6, 73–187 (1987)

    Article  Google Scholar 

  8. Bungartz, H.-J., Schäfer, M. (eds.): Fluid-Structure Interaction II. Modelling, Simulation, Optimisation. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2010)

    Google Scholar 

  9. Cottet, G.-H., Maitre, E., Mileent, T.: Eulerian formulation and level set models for incompressible fluid-structure interaction. Math. Model. Numer. Anal. 42, 471–492 (2008)

    Article  MATH  Google Scholar 

  10. Douglas, J., Russel, T.F.: Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dunne, T.: An Eulerian approach to fluid-structure interaction and goal-oriented mesh refinement. Int. J. Numer. Math. Fluids 51, 1017–1039 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dunne, T., Rannacher, R., Richter, T.: Numerical simulation of fluid-structure interaction based on monolithic variational formulations. In: Galdi, G.P., Rannacher, R. (eds.) Comtemporary Challenges in Mathematical Fluid Mechanics. World Scientific, Singapore (2010)

    Google Scholar 

  13. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. In: Iserles, A. (ed.) Acta Numerica 1995, pp. 105–158. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  14. Fernández, M.A., Gerbeau, J.-F.: Algorithms for fluid-structure interaction problems. In: Formaggia, L., Quarteroni, A., Veneziani, A. (eds.) Cardiovascular Mathematics. MS&A, vol. 1, pp. 307–346. Springer, Milan (2009)

    Google Scholar 

  15. Formaggia, L., Nobile, F.: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7, 105–132 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Formaggia, L., Nobile, F.: Stability analysis of second-order time accurate schemes for ALE-FEM. Comput. Methods Appl. Mech. Eng. 193(39–41), 4097–4116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frei, S., Richter, T.: Time-discretization of parabolic problems with moving interfaces. SIAM J. Numer. Anal. (2015, in preparation)

    Google Scholar 

  18. Frei, S., Richter, T.: A locally modified parametric finite element method for interface problems.SIAM J. Numer. Anal. 52(5), 2315–2334 (2014)

    Google Scholar 

  19. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 3523–3540 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, P., Qiao, R.: A full-Eulerian solid level set method for simulation of fluid-structure interactions. Microfluid Nanofluid 11, 557–567 (2011)

    Article  Google Scholar 

  23. Helenbrook, B.T.: Mesh deformation using the biharmonic operator. Int. J. Numer. Methods Eng. 56(7), 1007–1021 (2003)

    Article  MATH  Google Scholar 

  24. Hesse, H.K., Kanschat, G.: Mesh adaptive multiple shooting for partial differential equations. Part I: linear quadratic optimal control problems. J. Numer. Math. 17(3), 195–217 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. iii. smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489–512 (1988)

    Google Scholar 

  26. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. iv. error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(3), 353–384 (1990)

    Google Scholar 

  27. Hirt, C.W., Amsden, A.A., Cook, J.L.: An arbitrary lagrangian-eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–469 (1974)

    Article  MATH  Google Scholar 

  28. Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley-Blackwell, Engelska (2000)

    Google Scholar 

  29. Hron, J., Turek, S.: A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics. In: Bungartz, H.-J., Schäfer, M. (eds.) Fluid-Structure Interaction: Modeling, Simulation, Optimization. Lecture Notes in Computational Science and Engineering, pp. 146–170. Springer, Berlin (2006)

    Chapter  Google Scholar 

  30. Hron, J., Turek, S.: Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow. In: Bungartz, H.-J., Schäfer, M. (eds.) Fluid-Structure Interaction: Modeling, Simulation, Optimization. Lecture Notes in Computational Science and Engineering, pp. 371–385. Springer, Berlin (2006)

    Google Scholar 

  31. Hron, J., Turek, S., Madlik, M., Razzaq, M., Wobker, H., Acker, J.F.: Numerical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics. In: Bungartz, H.-J., Schäfer, M. (eds.) Fluid-Structure Interaction II: Modeling, Simulation, Optimization. Lecture Notes in Computational Science and Engineering, pp. 197–220. Springer, Berlin (2010)

    Google Scholar 

  32. Huerta, A., Liu, W.K.: Viscous flow with large free-surface motion. Comput. Methods Appl. Mech. Eng. 69(3), 277–324 (1988)

    Article  MATH  Google Scholar 

  33. Lions, J.-L., Mayday, Y., Turinici, G.: A parareal in time-discretization of PDEs. C. R. Acad. Sci. Paris Sér. I Math. 332, 661–668 (2001)

    Article  MATH  Google Scholar 

  34. Luskin, M., Rannacher, R.: On the smoothing propoerty of the Crank-Nicholson scheme. Appl. Anal. 14, 117–135 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  35. MacKinnon, R.J., Carey, G.F.: Treatment of material discontinuities in finite element computations. Int. J. Numer. Meth. Eng. 24, 393–417 (1987)

    Article  MATH  Google Scholar 

  36. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  37. Meidner, D., Richter, T.: A posteriori error estimation for the theta and fractional step theta time-stepping schemes. Comput. Methods Appl. Math. 14, 203–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Meidner, D., Richter, T.:A posteriori error estimation for the fractional step theta discretization of the incompressible Navier-Stokes equations.Comput. Methods Appl. Mech. Eng. 288, 45–59 (2015)

    Google Scholar 

  39. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  40. Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  41. Richter, T.: Fluid-structure interactions in Fully Eulerian coordinates. In: PAMM, 83st Annual Meeting of the International Association of Applied Mathematics and Mechanics, pp. 827–830 (2012)

    Google Scholar 

  42. Richter, T.: Goal oriented error estimation for fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 223–224, 28–42 (2012)

    Article  Google Scholar 

  43. Richter, T.: A Fully Eulerian formulation for fluid-structure-interaction problems. J. Comput. Phys. 233, 227–240 (2013)

    Article  MathSciNet  Google Scholar 

  44. Richter, T.: Finite Elements for Fluid-Structure Interactions. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2014, submitted)

    Google Scholar 

  45. Richter, T., Wick, T.: Finite elements for fluid-structure interaction in ALE and Fully Eulerian coordinates. Comput. Methods Appl. Mech. Eng. (2010). doi:10.1016/j.cma.2010.04.016

    MathSciNet  Google Scholar 

  46. Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder. (With support by F. Durst, E. Krause and R. Rannacher). In: Hirschel, E.H. (ed.) Flow Simulation with High-Performance Computers II. DFG priority research program results 1993–1995. Notes Numer. Fluid Mech., vol. 52, pp. 547–566. Vieweg, Wiesbaden (1996)

    Google Scholar 

  47. Sethian, J.A.: Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry. Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  48. Shubin, G.R., Bell, J.B.: An analysis of grid orientation effect in numerical simulation of miscible displacement. Comput. Methods Appl. Mech. Eng. 47, 47–71 (1984)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  50. Sugiyama, K., Li, S., Takeuchi, S., Takagi, S., Matsumato, Y.: A full Eulerian finite difference approach for solving fluid-structure interacion. J. Comput. Phys. 230, 596–627 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Takagi, S., Sugiyama, K., Matsumato, Y.: A review of full Eulerian mehtods for fluid structure interaction problems. J. Appl. Mech. 79(1), 010,911 (2012)

    Google Scholar 

  52. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (1997)

    Google Scholar 

  53. Tikhonov, A.N., Samarskii, A.A.: Homogeneous difference schemes. USSR Comput. Math. Math. Phys. 1, 5–67 (1962)

    Article  Google Scholar 

  54. Turek, S., Rivkind, L., Hron, J., Glowinski, R.: Numerical analysis of a new time-stepping theta-scheme for incompressible flow simulations. Ergebnisberichte des Instituts für Angewandte Mathematik, vol. 282. Technical Report, Fakultät für Mathematik, TU Dortmund (2005).

    Google Scholar 

  55. Turek, S., Hron, J., Razzaq, M., Wobker, H., Schäfer, M.: Numerical benchmarking of fluid-structure interaction: a comparison of different discretization and solution approaches. In: Bungartz, H.J., Mehl, M., Schäfer, M. (eds.) Fluid Structure Interaction II: Modeling, Simulation and Optimization. Springer, Berlin (2010)

    Google Scholar 

  56. Wick, T.: Adaptive finite element simulation of fluid-structure interaction with application to heart-valve dynamics. Ph.D. thesis, Univeristy of Heidelberg (2011). urn:nbn:de:bsz:16-opus-129926

    Google Scholar 

  57. Wick, T.: Fluid-structure interactions using different mesh motion techniques. Comput. Struct. 89(13–14), 1456–1467 (2011)

    Article  Google Scholar 

  58. Wick, T.: Coupling of fully Eulerian and arbitrary Lagrangian-Eulerian methods for fluid-structure interaction computations problems. Comput. Mech. 52(5), 1113–1124 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wick, T.: Coupling of fully Eulerian and arbitrary Lagrangian-Eulerian methods for fluid-structure interaction computations. Comput. Mech. 52(5), 1113–1124 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  60. Wick, T.: Flapping and contact FSI computations with the fluid—solid Interface-tracking/Interface-capturing technique and mesh adaptivity. Comput. Mech. 53(1), 29–43 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wick, T.: Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal.ii library. Arch. Numer. Softw. 1, 1–19 (2013)

    Google Scholar 

  62. Wick, T.: Stability estimates and numerical comparison of second order time-stepping schemes for fluid-structure interactions. In: Cangiani, A., Davidchack, R.L., Georgoulis, E., Gorban, A.N., Levesley, J., Tretyakov, M.V. (eds.) Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011, Leicester, September 2011, pp. 625–632 (2013)

    MathSciNet  Google Scholar 

  63. Xie, H., Ito, K., Li, Z.-L., Toivanen, J.: A finite element method for interface problems with locally modified triangulation. Contemp. Math. 466, 179–190 (2008)

    Article  MathSciNet  Google Scholar 

  64. Zhao, H., Freund, J.B., Moser, R.D.: A fixed-mesh method for incompressible flow-structure systems with finite solid deformations. J. Comput. Phys. 227(6), 3114–1340 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, INF 294, 69120, Heidelberg, Germany

    Thomas Richter

  2. The Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Thomas Wick

Authors
  1. Thomas Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Wick
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Richter .

Editor information

Editors and Affiliations

  1. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Thomas Carraro

  2. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Michael Geiger

  3. Heidelberg University, Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

    Stefan Körkel

  4. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Rolf Rannacher

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Richter, T., Wick, T. (2015). On Time Discretizations of Fluid-Structure Interactions. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation