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Chinese Annals of Mathematics, Series B

, Volume 39, Issue 2, pp 213–232 | Cite as

An Energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme

  • Olivier Pironneau
Article

Abstract

The conservation laws of continuum mechanics, written in an Eulerian frame, do not distinguish fluids and solids, except in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown monolithic methods for fluid structure interactions (FSI for short) are built. In this paper such a formulation is analysed when the solid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galerkin method render this approach feasible and reasonably robust. In this paper the method and its discretisation are presented, stability is discussed through an energy estimate. A numerical section discusses implementation issues and presents a few simple tests.

Keywords

Fluid-Structure interactions Numerical method Energy stability Finite element method 

2000 MR Subject Classification

65M60 74F10 74S30 76D05 76M25 

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Notes

Acknowledgements

The author thanks Frédéric Hecht for very valuable discussions and comments.

References

  1. [1]
    Antman, S. S., Nonlinear Problems of Elasticity, (2nd ed.), Applied Mathematical Sciences, 107, Springer-Verlag, New York, 2005.zbMATHGoogle Scholar
  2. [2]
    Bathe, K. J., Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996.zbMATHGoogle Scholar
  3. [3]
    Bathe, K. J., Ramm, E. and Wilson, E. L., Finite element formulations for large deformation dynamic analysis, Int. J. Numer. Methods Eng., 9(2), 1975, 353–386.CrossRefzbMATHGoogle Scholar
  4. [4]
    Boffi, D., Brezzi, F. and Fortin, M., Mixed Finite Element Methods and Applications, Computational Mathematics, Heidelberg, 44, Springer-Verlag, Berlin, 2013.CrossRefzbMATHGoogle Scholar
  5. [5]
    Boffi, D., Cavallini, N. and Gastaldi, L., The finite element immersed boundary method with distributed Lagrange multiplier, SIAM J. Numer. Anal., 53(6), 2015, 2584–2604.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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(12), 2003, 985–990.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bukaca, M., Canic, S., Glowinski, R., et al., Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement, Journal of Computational Physics, 235, 2013, 515–541.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Chiang, C.-Y., Pironneau, O., Sheu, T. and Thiriet, M., Numerical study of a 3D Eulerian monolithic formulation for fluid-structure-interaction, Fluids, 2017.Google Scholar
  9. [9]
    Ciarlet, P. G., Mathematical Elasticity, I., Three-dimensional Elasticity, North Holland, Amsterdam, 1988.zbMATHGoogle Scholar
  10. [10]
    Cottet, G. H., Maitre, E. and Milcent, T., Eulerian formulation and level set models for incompressible fluid-structure interaction, M2AN Math. Model. Numer. Anal., 42(3), 2008, 471–492.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Coupez, T., Silva, L. and Hachem, E., Implicit Boundary and Adaptive Anisotropic Meshes, New challenges in Grid Generation and Adaptivity for Scientific Computing, S. Peretto and L. Formaggia (eds.), 5, Springer-Verlag, Cham, 2015.Google Scholar
  12. [12]
    Coutand, D. and Shkoller, S., Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176(1), 2005, 25–102.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Dunne, T., Adaptive finite element approximation of fluid-structure interaction based on an Eulerian variational formulation, ECCOMAS CFD, 2006, Wesseling, P., O˜nate, E. and Périaux, J. (eds.), Elsevier, TU Delft, The Netherlands, 2006.CrossRefzbMATHGoogle Scholar
  14. [14]
    Dunne, T., An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaptation, Int. J. Numer. Meth. Fluids, 51, 2006, 1017–1039.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Dunne, Th. and Rannacher, R. Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on an Eulerian Variational Formulation, Fluid-Structure Interaction: Modelling, Simulation, Optimization, Bungartz, H-J. and Schaefer, M. (eds.), Lecture Notes in Computational Science and Engineering, 53, Springer-Verlag, Berlin, 2006, 110–146.CrossRefGoogle Scholar
  16. [16]
    Fernandez, M. A., Mullaert, J. and Vidrascu, M., Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures, Comp. Methods in Applied Mech. and Engg., 267, 2013, 566–593.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Formaggia, L., Quarteroni, A. and Veneziani, A., Alessandro Multiscale Models of the Vascular System, Cardiovasuclar Mathematics, Springer-Verlag, Italia, Milan, 2009, 395–446.Google Scholar
  18. [18]
    Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 2012, 251–265, http://www.FreeFem. org.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Hecht, F. and Pironneau, O., An energy stable monolithic Eulerian fluid-structure finite element method, International Journal for Numerical Methods in Fluids, 85(7), 2017, 430–446.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Change Heil, Matthias to Heil, M., Solvers for large-displacement fluid structure interaction problems: Segregated versus monolithic approaches, Comput. Mech., 43, 2008, 91–101.CrossRefzbMATHGoogle Scholar
  21. [21]
    Hron, J. and Turek, S., A monolithic fem solver for an ALE formulation of fluid-structure interaction with configuration for numerical benchmarking, European Conference on Computational Fluid Dynamics ECCOMAS CFD, 2006, Wesseling, P., Onate, E. and Periaux, J. (eds.), TU Delft, The Netherlands, 2006.Google Scholar
  22. [22]
    Léger, S., Méthode lagrangienne actualisée pour des problèmes hyperélastiques en très grandes déformations, Thèse de Doctorat, Université Laval, 2014 (in France).Google Scholar
  23. [23]
    Le Tallec, P. and Hauret, P., Energy conservation in Fluid-Structure Interactions, Numerical Methods for Scientific Computing, Variational Problems And Applications, Neittanmaki, P., Kuznetsov, Y. and Pironneau, O. (eds.), CIMNE, Barcelona, 2003.Google Scholar
  24. [24]
    Le Tallec, P. and Mouro, J., Fluid structure interaction with large structural displacements, Comp. Meth. Appl. Mech. Eng., 190(24–25), 2001, 3039–3068.CrossRefzbMATHGoogle Scholar
  25. [25]
    Liu, J., A second-order changing-connectivity ALE scheme and its application to FSI with large convection of fluids and near-contact of structures, Journal of Computational Physics, 304, 2016, 380–423.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Liu, I-Shih, Cipolatti, R. and Rincon, M. A., Incremental Linear Approximation for Finite Elasticity, Proc. ICNAAM, Wiley, 2006.zbMATHGoogle Scholar
  27. [27]
    Marsden, J. and Hughes, T. J. R., Mathematical Foundations of Elasticity, Dover Publications, New York, 1994.zbMATHGoogle Scholar
  28. [28]
    Nobile, F. and Vergara, C., An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions, SIAM J. Sci. Comp., 30(2), 2008, 731–763.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Peskin, C. S., The immersed boundary method, Acta Numerica, 11, 2002, 479–517.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Pironneau, O., Numerical Study of a Monolithic Fluid-Structure Formulation, Variational Analysis and Aerospace Engineering, 116, Springer-Verlag, Cham, 2016.Google Scholar
  31. [31]
    Rannacher, R. and Richter, T., An Adaptive Finite Element Method for Fluid-Structure Interaction Problems Based on a Fully Eulerian Formulation, Lecture Notes in Computational Science and Engineering, 73, Springer-Verlag, Heidelberg, 2010.zbMATHGoogle Scholar
  32. [32]
    Raymond, J.-P. and Vanninathan, M., A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl., 102, 2014, 546–596.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Richter, Th. and Wick, Th., Finite elements for fluid-structure interaction in ALE and fully Eulerian coordinates, Comput. Methods Appl. Mech. Engrg., 199, 2010, 2633–2642.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Wang, Y. X., The Accurate and Efficient Numerical Simulation of General Fluid Structure Interaction: A Unified Finite Element Method, Proc. Conf. on FSI problems, IMS-NUS, Singapore, 2016.Google Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UPMC (Paris VI) Laboratoire Jacques-Louis Lions Place JussieuSorbonne UniversitéParisFrance

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