Skip to main content
SpringerLink
Log in

A Monolithic Approach to Fluid–Composite Structure Interaction

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We study a nonlinear fluid–structure interaction (FSI) problem between an incompressible, viscous fluid and a composite elastic structure consisting of two layers: a thin layer (membrane) in direct contact with the fluid, and a thick layer (3D linearly elastic structure) sitting on top of the thin layer. The coupling between the fluid and structure, and the coupling between the two structures is achieved via the kinematic and dynamic coupling conditions modeling no-slip and balance of forces, respectively. The coupling is evaluated at the moving fluid–structure interface with mass, i.e., the thin structure. To solve this nonlinear moving-boundary problem in 3D, a monolithic, fully implicit method was developed, and combined with an arbitrary Lagrangian–Eulerian approach to deal with the motion of the fluid domain. This class of problems and its generalizations are important in e.g., modeling FSI between blood flow and arterial walls, which are known to be composed of several different layers, each with different mechanical characteristics and thickness. By using this model we show how multi-layered structure of arterial walls influences the pressure wave propagation in arterial walls, and how the presence of atheroma and the presence of a vascular device called stent, influence intramural strain distribution throughout different layers of the arterial wall. The detailed intramural strain distribution provided by this model can be used in conjunction with ultrasound B-mode scans as a predictive tool for an early detection of atherosclerosis (Zahnd et al. in IEEE international on ultrasonics symposium (IUS), pp 1770–1773, 2011).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Alfonso, F., Byrne, R.A., Rivero, F., Kastrati, A.: Current treatment of in-stent restenosis (state-of-the-art paper). J. Am. Coll. Cardiol. 63(24), 2659–2673 (2014)

    Article  Google Scholar 

  2. Baaijens, F.P.T.: A fictitious domain/mortar element method for fluid–structure interaction. Int. J. Numer. Methods Fluids 35(7), 743–761 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Badia, S., Nobile, F., Vergara, C.: Fluid–structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227, 7027–7051 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Badia, S., Quaini, A., Quarteroni, A.: Modular vs. non-modular preconditioners for fluid–structure systems with large added-mass effect. Comput. Methods Appl. Mech. Eng. 197(49–50), 4216–4232 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balzani, D., Deparis, S., Fausten, S., Forti, D., Heinlein, A., Klawonn, A., Quarteroni, A., Rheinbach, O., Schröder, J.: Numerical modeling of fluid–structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains. Int. J. Numer. Methods Biomed. Eng. doi:10.1002/cnm.2756

  6. Barker, A.T., Cai, X.-C.: Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling. J. Comput. Phys. 229(3), 642–659 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bazilevs, Y., Calo, V.M., Zhang, Y., Hughes, T.J.R.: Isogeometric fluid–structure interaction: theory, algorithms and computations. Comput. Mech. 43(1), 3–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berry, J.L., Santamarina, A., Moore Jr., J.E., Roychowdhury, S., Routh, W.D.: Experimental and computational flow evaluation of coronary stents. Ann. Biomed. Eng. 28, 386–398 (2000)

    Article  Google Scholar 

  9. Bonito, A., Nochetto, R., Pauletti, M.: Dynamics of biomembranes: effect of the bulk fluid. Math. Model. Nat. Phenom. 6(05), 25–43 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bukac, M., Canic, S., Muha, B.: A partitioned scheme for fluid–composite structure interaction problems. J. Comput. Phys. 281, 493–517 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Canic, S., Kim, E.H.: Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axisymmetric vessels. Math. Methods Appl. Sci. 26(14), 1161–1186 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Causin, P., Gerbeau, J.F., Nobile, F.: Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Comput. Methods Appl. Mech. Eng. 194(42–44), 4506–4527 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colciago, C., Deparis, S., Quarteroni, A.: Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics. J. Comput. Appl. Math. 265, 120–138 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Crosetto, P., Deparis, S., Fourestey, G., Quarteroni, A.: Parallel algorithms for fluid–structure interaction problems in haemodynamics. SIAM J. Sci. Comput. 33(4), 1598–1622 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crosetto, P., Deparis, S., Fourestey, G., Quarteroni, A.: Parallel algorithms for fluid–structure interaction problems in haemodynamics. SIAM J. Sci. Comput. 33(4), 1598–1622 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deparis, S., Forti, D., Grandperrin, G., Quarteroni, A.: FaCSI: a block parallel preconditioner for fluid–structure interaction in hemodynamics. J. Comput. Phys. 327, 700–718 (2016)

    Article  MathSciNet  Google Scholar 

  18. Dettmer, W.G., Perić, D.: A fully implicit computational strategy for strongly coupled fluid–solid interaction. Arch. Comput. Methods Eng. 14(3), 205–247 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Donea, J.: Arbitrary Lagrangian–Eulerian Finite Element Methods, in: Computational Methods for Transient Analysis. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  20. Dumoulin, C., Cochelin, B.: Mechanical behaviour modelling of balloon-expandable stents. J. Biomech. 33(11), 1461–1470 (2000)

    Article  Google Scholar 

  21. Fang, H., Wang, Z., Lin, Z., Liu, M.: Lattice Boltzmann method for simulating the viscous flow in large distensible blood vessels. Phys. Rev. E 65(5), 051925-1–051925-12 (2002)

    Article  Google Scholar 

  22. Fauci, L.J., Dillon, R.: Biofluidmechanics of reproduction. In: Annual Review of Fluid Mechanics. Vol. 38 of Annu. Rev. Fluid Mech., pp. 371–394. Annual Reviews, Palo Alto (2006)

  23. Feng, Z.G., Michaelides, E.E.: The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J. Comput. Phys. 195(2), 602–628 (2004)

    Article  MATH  Google Scholar 

  24. Figueroa, A., Vignon-Clementel, I.E., Jansen, K., Hughes, T., Taylor, C.: A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput. Methods Appl. Mech. Eng. 195(41–43), 5685–5706 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Figueroa, A., Baek, S., Taylor, C., Humphrey, J.: A computational framework for fluid–solid-growth modeling in cardiovascular simulations. Comput. Methods Appl. Mech. Eng. 198(45), 3583–3602 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fogelson, A.L., Guy, R.D.: Platelet-wall interactions in continuum models of platelet thrombosis: formulation and numerical solution. Math. Med. Biol. 21, 293–334 (2004)

    Article  MATH  Google Scholar 

  27. Frank, A.O., Walsh, P.W., Moore Jr., J.E.: Computational fluid dynamics and stent design. Artif. Organs 26(7), 614–621 (2002)

    Article  Google Scholar 

  28. Gee, M.W., Küttler, U., Wall, W.A.: Truly monolithic algebraic multigrid for fluid–structure interaction. Int. J. Numer. Methods Eng. 85(8), 987–1016 (2011)

    Article  MATH  Google Scholar 

  29. Griffith, B.E.: Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int. J. Numer. Methods Biomed. Eng. 28(3), 317–345 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Griffith, B.E.: On the volume conservation of the immersed boundary method. Commun. Comput. Phys. 12(2), 401–432 (2012)

    Article  MathSciNet  Google Scholar 

  31. Griffith, B.E., Hornung, R.D., McQueen, D.M., Peskin, C.S.: An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys. 223(1), 10–49 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Griffith, B.E., Luo, R., McQueen, D.M., Peskin, C.S.: Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method. Int. J. Appl. Mech. 1, 137–177 (2009)

    Article  Google Scholar 

  33. Heil, M.: An efficient solver for the fully coupled solution of large-displacement fluid–structure interaction problems. Comput. Methods Appl. Mech. Eng. 193(1–2), 1–23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hughes, T.J.R., Liu, W.K., Zimmermann, T.K.: Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  35. Krafczyk, M., Cerrolaza, M., Schulz, M., Rank, E.: Analysis of 3D transient blood flow passing through an artificial aortic valve by lattice-Boltzmann methods. J. Biomech. 31(5), 453–462 (1998)

    Article  Google Scholar 

  36. Krafczyk, M., Tölke, J., Rank, E., Schulz, M.: Two-dimensional simulation of fluid–structure interaction using lattice-Boltzmann methods. Comput. Struct. 79(22–25), 2031–2037 (2001)

    Article  Google Scholar 

  37. Lau, K.W., Johan, A., Sigwart, U., Hung, J.S.: A stent is not just a stent: stent construction and design do matter in its clinical performance. Singap. Med. J. 45(7), 305–312 (2004)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  39. Leuprecht, A., Perktold, K., Prosi, M., Berk, T., Trubel, W., Schima, H.: Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts. J. Biomech. 35(2), 225–236 (2002)

    Article  Google Scholar 

  40. Lim, S., Peskin, C.S.: Simulations of the whirling instability by the immersed boundary method. SIAM J. Sci. Comput. 25(6), 2066–2083 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Marques, K.M., Spruijt, H.J., Boer, C., Westerhof, N., Visser, C.A., Visser, F.C.: The diastolic flow-pressure gradient relation in coronary stenoses in humans. J. Am. Coll. Cardiol. 39, 1630–1635 (2002)

  42. McClean, D.R., Eigler, N.: Stent design: implications for restenosis. MedReviews, LLC (2002)

  43. Migliavacca, F., Petrini, L., Colombo, M., Auricchio, F., Pietrabissa, R.: Mechanical behavior of coronary stents investigated through the finite element method. J. Biomech. 35(6), 803–811 (2002)

    Article  Google Scholar 

  44. Miller, L.A., Peskin, C.S.: A computational fluid dynamics of ’clap and fling’ in the smallest insects. J. Exp. Biol. 208(2), 195–212 (2005)

    Article  Google Scholar 

  45. Moore Jr., J.E., Berry, J.L.: Fluid and solid mechanical implications of vascular stenting. Ann. Biomed. Eng. 30(4), 498–508 (2002)

    Article  Google Scholar 

  46. Morton, A.C., Crossman, D., Gunn, J.: The influence of physical stent parameters upon restenosis. Pathol. Biol. (Paris) 52(4), 196–205 (2004)

    Article  Google Scholar 

  47. Muha, B.: A note on optimal regularity and regularizing effects of point mass coupling for a heat-wave system. J. Math. Anal. Appl. 425(2), 1134–1147 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Muha, B., Canic, S.: Existence of a weak solution to a nonlinear fluid–structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch. Ration. Mech. Anal. 207(3), 919–968 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Muha, B., Canic, S.: existence of a solution to a fluid-multi-layered–structure interaction problem. J. Differ. Equ. 256, 658–706 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nobile, F.: Numerical approximation of fluid–structure interaction problems with application to haemodynamics. Ph.D. Thesis, EPFL (2001)

  51. Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  52. Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  53. Quaini, A., Quarteroni, A.: A semi-implicit approach for fluid–structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci. 17(6), 957–985 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Quarteroni, A., Tuveri, M., Veneziani, A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2(4), 163–197 (2000)

    Article  MATH  Google Scholar 

  55. Tambaca, J., Canic, S., Kosor, M., Fish, R.D., Paniagua, D.: Mechanical behavior of fully expanded commercially available endovascular coronary stents. Tex. Heart Inst. J. 38, 491–501 (2011)

    Google Scholar 

  56. Tezduyar, T.E., Sathe, S., Stein, K.: Solution techniques for the fully discretized equations in computation of fluid–structure interactions with the space time formulations. Comput. Methods Appl. Mech. Eng. 195(4143), 5743–5753 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  57. Timmins, L.H., Moreno, M.R., Meyer, C.A., Criscione, J.C., Rachev, J.E., Moore Jr., A.: Stented artery biomechanics and device design optimization. Med. Biol. Eng. Comput. 45(5), 505–513 (2007)

    Article  Google Scholar 

  58. Van Loon, R., Anderson, P.D., De Hart, J., Baaijens, F.P.T.: A combined fictitious domain/adaptive meshing method for fluid–structure interaction in heart valves. Int. J. Numer. Methods Fluids 46(5), 533–544 (2004)

    Article  MATH  Google Scholar 

  59. Wu, Y., Cai, X.-C.: A fully implicit domain decomposition based ALE framework for three-dimensional fluid–structure interaction with application in blood flow computation. J. Comput. Phys. 258, 524–537 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zahnd, G., Boussel, L., Serusclat, A., Vray, D.: Intramural shear strain can highlight the presence of atherosclerosis: a clinical in vivo study. In: 2011 IEEE International on Ultrasonics Symposium (IUS), pp. 1770–1773 (2011)

Download references

Acknowledgements

All the authors would like to thank Professor Alfio Quarteroni for his support of this research and of D. Forti’s visit to UH. Additionally, the research of D. Forti was supported by the Swiss National Foundation (SNF), Project No. 140184. S. Deparis and D. Forti gratefully acknowledge the Swiss National Supercomputing Center (CSCS) for providing the CPU resources for the numerical simulations under Project ID s475. The research of S. Canic and M. Bukac was supported by the US National Science Foundation under Grant DMS-1318763, which also provided partial support for D. Forti’s visit to the University of Houston. Additionally, the research of M. Bukac was supported by the US National Science Foundation under Grant DMS-1619993. The research of S. Canic and A. Quaini was supported by the US National Science Foundation under Grant DMS-1263572. Additionally, the research of A. Quaini was supported by the US National Science Foundation under Grant DMS-1620384. Additionally, the research of S. Canic was supported by NSF DMS-1311709 and by the University of Houston Hugh Roy and Lillie Cranz Cullen Distinguished Professorship funds, which also provided additional travel support for D. Forti’s visit to UH. The authors acknowledge the use of the Maxwell Cluster and the advanced support from the Center of Advanced Computing and Data Systems at the University of Houston to carry out the research presented here.

Author information

Authors and Affiliations

  1. CMCS - Chair of Modeling and Scientific Computing, MATHICSE - Mathematics Institute of Computational Science and Engineering, EPFL - École Polytechnique Fédérale de Lausanne, Station 8, Lausanne, 1015, Switzerland

    Davide Forti & Simone Deparis

  2. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 158 Hurley Hall, Notre Dame, IN, 46556, USA

    Martina Bukac

  3. Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX, 77204, USA

    Annalisa Quaini & Suncica Canic

Authors
  1. Davide Forti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martina Bukac
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Annalisa Quaini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Suncica Canic
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Simone Deparis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annalisa Quaini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Forti, D., Bukac, M., Quaini, A. et al. A Monolithic Approach to Fluid–Composite Structure Interaction. J Sci Comput 72, 396–421 (2017). https://doi.org/10.1007/s10915-017-0363-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0363-5

Keywords

Search

Navigation