On the Differentiability of Fluid–Structure Interaction Problems with Respect to the Problem Data


A coupled system of stationary fluid–structure equations in an arbitrary Lagrangian–Eulerian framework is considered in this work. Existence results presented in the literature are extended to show differentiability of the solutions to a stationary fluid–structure interaction problem with respect to the given data, volume forces and boundary values, provided a small data assumption holds. Numerical experiments are used to substantiate the theoretical findings.

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  1. 1.

    Allen, M., Maute, K.: Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena. Comput. Methods Appl. Mech. Eng. 194, 3472–3495 (2005)

    ADS  Article  Google Scholar 

  2. 2.

    Avalos, G., Lasiecka, I., Triggiani, R.: Higher regularity of a coupled parabolic–hyperbolic fluid–structure interactive system. Georgian Math. J. 15, 403–437 (2010)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33, 24/1–24/27 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations, 1st edn. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2003)

    Google Scholar 

  5. 5.

    Barbu, V., Grujic, Z., Lasiecka, I., Tuffaha, A.: Smoothness of weak solutions to a nonlinear fluid–structure interaction model. Indiana Univ. Math. J. 57, 1173–1207 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bazilevs, Y., Takizawa, K., Tezduyar, T.: Computational Fluid–Structure Interaction: Methods and Applications. Wiley, London (2013)

    Google Scholar 

  7. 7.

    Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation. Acta Numer. 2001, 1–102 (2001)

    Article  Google Scholar 

  9. 9.

    Bodnar, T., Galdi, G., Necasova, S.: Fluid–Structure Interaction and Biomedical Applications. Birkhaeuser, Basel (2014)

    Google Scholar 

  10. 10.

    Boulakia, M.: Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid. C. R. Math. Acad. Sci. Paris 336, 985–990 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bucci, F., Lasiecka, I.: Optimal boundary control with critical penalization for a pde model of fluid–solid interactions. Calc. Var. Partial Differ. Equ. 37, 217–235 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bungartz, H.-J., Schäfer, M. (eds.): Fluid–Structure Interaction: Modelling, Simulation, Optimisation, vol. 53 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2006)

  13. 13.

    Ciarlet, P.G.: Mathematical Elasticity. Volume 1: Three Dimensional Elasticity. North-Holland, Amsterdam (1984)

    Google Scholar 

  14. 14.

    Coutand, D., Shkoller, S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Ration. Mech. Anal. 176, 25–102 (2005)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Coutand, D., Shkoller, S.: The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Ration. Mech. Anal. 179, 303–352 (2006)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Deneuvy, A.C.: Theoretical study and optimization of a fluid–structure interaction problem. M2AN Math. Model. Numer. Anal. 22, 75–92 (1988)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Desjardins, B., Esteban, M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146, 59–71 (1999)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Desjardins, B., Esteban, M.J., Grandmont, C., Le Tallec, P.: Weak solutions for a fluid–structure interaction problem. Rev. Mat. Complut. 14, 523–538 (2001)

    MathSciNet  Article  Google Scholar 

  19. 19.

    dos Santos, N.D., Gerbeau, J.-F., Bourgat, J.F.: A partitioned fluid–structure algorithm for elastic thin valves with contact. Comput. Methods Appl. Mech. Eng. 197, 1750–1761 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Dunne, T., Richter, T., Rannacher, R.: Numerical simulation of fluid–structure interaction based on monolithic variational formulations. In: Comtemporary Challenges in Mathematical Fluid Mechanics, pp. 1–75. World Scientific, Singapore (2010)

    Google Scholar 

  21. 21.

    Failer, L., Meidner, D., Vexler, B.: Optimal control of a linear unsteady fluid–structure interaction problem. J. Optim. Theory Appl. 170, 1–27 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Failer, L., Wick, T.: Adaptive time-step control for nonlinear fluid–structure interaction. J. Comput. Phys. 366, 448–477 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    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 

  24. 24.

    Formaggia, L., Quarteroni, A., Veneziani, A. (eds.): Cardiovascular Mathematics, vol. 1 of MS&A. Modeling, Simulation and Applications. Modeling and Simulation of the Circulatory System. Springer, Milan (2009)

  25. 25.

    Frei, S., Holm, B., Richter, T., Wick, T., Yang, H.: Fluid–Structure Interaction: Modeling, Adaptive Discretisations and Solvers. Walter de Gruyter, Berlin (2017)

    Google Scholar 

  26. 26.

    Galdi, G., Rannacher, R.: Fundamental Trends in Fluid–Structure Interaction. World Scientific, Singapore (2010)

    Google Scholar 

  27. 27.

    Goll, C., Wick, T., Wollner, W.: DOpElib: differential equations and optimization environment; a goal oriented software library for solving PDEs and optimization problems with PDEs. Arch. Numer. Softw. 5, 1–14 (2017)

    Google Scholar 

  28. 28.

    Grandmont, C.: Existence et unicité de solutions d’un problème de couplage fluide-structure bidimensionnel stationnaire. C. R. Acad. Sci. Paris 326, 651–656 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Grandmont, C.: Existence for a three-dimensional steady state fluid–structure interaction problem. J. Math. Fluid Mech. 4, 76–94 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Grandmont, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40, 716–737 (2008)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Grätsch, T., Bathe, K.-J.: Goal-oriented error estimation in the analysis of fluid flows with structural interactions. Comput. Methods Appl. Mech. Eng. 195, 5673–5684 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Helenbrook, B.: Mesh deformation using the biharmonic operator. Int. J. Numer. Methods Eng. 56, 1–30 (2001)

    Google Scholar 

  33. 33.

    Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  34. 34.

    Hirth, C., Amsden, A., Cook, J.: An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253 (1974)

    ADS  Article  Google Scholar 

  35. 35.

    Hron, J., Turek, S.: Proposal for Numerical Benchmarking of Fluid–Structure Interaction Between an Elastic Object and Laminar Incompressible Flow, vol. 53, pp. 146–170. Springer, Berlin (2006)

    Google Scholar 

  36. 36.

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

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Langer, U., Yang, H.: Partitioned solution algorithms for fluid–structure interaction problems with hyperelastic models. J. Comput. Appl. Math. 276, 47–61 (2015)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Lund, E., Moller, H., Jakobsen, L.A.: Shape design optimization of stationary fluid–structure interaction problems with large displacements and turbulence. Struct. Multidiscip. Optim. 25, 383–392 (2003)

    Article  Google Scholar 

  39. 39.

    Murea, C.M., Vázquez, C.: Sensitivity and approximation of coupled fluid–structure equations by virtual control method. Appl. Math. Optim. 52, 183–218 (2005)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Nobile, F.: Numerical Approximation of Fluid–Structure Interaction Problems with Applications to Haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2001)

  41. 41.

    Nobile, F., Vergara, C.: An effective fluid–structure interaction formulation for vascular dynamics by generalized robin conditions. SIAM J. Sci. Comput. 30, 731–763 (2008)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Noh, W.: A Time-Dependent Two-Space-Dimensional Coupled Eulerian-Lagrangian Code, Vol. 3 of Methods Comput. Phys, pp. 117–179. Academic Press, New York (1964)

    Google Scholar 

  43. 43.

    Perego, M., Veneziani, A., Vergara, C.: A variational approach for estimating the compliance of the cardiovascular tissue: an inverse fluid–structure interaction problem. SIAM J. Sci. Comput. 33, 1181–1211 (2011)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Piperno, S., Farhat, C.: Paritioned procedures for the transient solution of coupled aeroelastic problems—part ii: energy transfer analysis and three-dimensional applications. Comput. Methods Appl. Mech. Eng. 190, 3147–3170 (2001)

    ADS  Article  Google Scholar 

  45. 45.

    Quaini, A., Canic, S., Glowinski, R., Igo, S., Hartley, C.J., Zoghbi, W., Little, S.: Validation of a 3d comoputational fluid–structure interaction model simulating flow through elastic aperature. J. Biomech. 45, 310–318 (2012)

    Article  Google Scholar 

  46. 46.

    Raymond, J.-P.: Feedback stabilization of a fluid–structure model. SIAM J. Control Optim. 48, 5398–5443 (2010)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Richter, T.: Goal-oriented error estimation for fluid–structure interaction problems. Comput. Methods Appl. Mech. Eng. 223–224, 38–42 (2012)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Richter, T.: Fluid–Structure Interactions: Models, Analysis, and Finite Elements. Springer, Berlin (2017)

    Google Scholar 

  49. 49.

    Richter, T., Wick, T.: Optimal control and parameter estimation for stationary fluid–structure interaction problems. SIAM J. Sci. Comput. 35, B1085–B1104 (2013)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Schäfer, M., Sternel, D., Becker, G., Pironkov, P.: Efficient Numerical Simulation and Optimization of Fluid–Structure Interaction, vol. 53, pp. 133–160. Springer, Berlin (2010)

    Google Scholar 

  51. 51.

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

    ADS  Article  Google Scholar 

  52. 52.

    Takizawa, K., Tezduyar, T.: Computational methods for parachute fluid–structure interactions. Arch. Comput. Methods Eng. 19, 125–169 (2012)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1977)

    Google Scholar 

  54. 54.

    Turek, S., Hron, J., Razzaq, M., Wobker, H., Schäfer, M.: Numerical Benchmarking of Fluid–Structure Interaction: A Comparison of Different Discretization and Solution Approaches, Fluid Structure Interaction II: Modelling, Simulation, Optimization, pp. 413–424. Springer, Heidelberg (2010)

    Google Scholar 

  55. 55.

    van der Zee, K., van Brummelen, E., de Borst, R.: Goal-oriented error estimation for Stokes flow interacting with a flexible channel. Int. J. Numer. Methods Fluids 56, 1551–1557 (2008)

    MathSciNet  Article  Google Scholar 

  56. 56.

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

    Article  Google Scholar 

  57. 57.

    Wick, T.: Goal-oriented mesh adaptivity for fluid–structure interaction with application to heart-valve settings. Arch. Mech. Eng. 59, 73–99 (2012)

    Article  Google Scholar 

  58. 58.

    Winslow, A.: Numerical solution of the quasi-linear poisson equation in a nonuniform triangle mesh. J. Comput. Phys. 1, 149–172 (1967)

    ADS  Article  Google Scholar 

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Wick, T., Wollner, W. On the Differentiability of Fluid–Structure Interaction Problems with Respect to the Problem Data. J. Math. Fluid Mech. 21, 34 (2019). https://doi.org/10.1007/s00021-019-0439-0

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Mathematics Subject Classification

  • Primary 74F10
  • Secondary 35Q35


  • Fluid–structure interactions
  • Differentiability of solutions with respect to problem data