Abstract
Mathematical problems involving the coupling of an incompressible viscous flow with an elastic structure appear in a large variety of engineering fields (see, e.g., [14, 17, 19–21]). This problem is considered here within a heterogenous domain decomposition framework, with the aim of using independent well-suited solvers for the fluid and the solid. One of the main difficulties that have to be faced under this approach is that the coupling can be very stiff. In particular, traditional Dirichlet-Neumann explicit coupling methods, which solve for the fluid (Dirichlet) and for the solid (Neumann) only once per time-step, are unconditionally unstable whenever the amount of added-mass effect in the system is large (see, e.g., [5, 12]). Typically this happens when the fluid and solid densities are close and the fluid domain is slender, as in hemodynamical applications. This explains, in part, the tremendous amount of work devoted over the last decade to the development of alternative coupling paradigms (see, e.g., [7] for a review).
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References
S. Badia, F. Nobile, C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227(14), 7027–7051 (2008)
M. Bukač, S. Čanić, R. Glowinski, J. Tambača, A. Quaini, Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement. J. Comput. Phys. 235, 515–541 (2013)
E. Burman, M.A. Fernández, Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Eng. 198(5–8), 766–784 (2009)
E. Burman, M.A. Fernández, Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling. Int. J. Numer. Methods Eng. 97(10), 739–758 (2014)
P. Causin, J.F. Gerbeau, F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194(42–44), 4506–4527 (2005)
D. Chapelle, K.J. Bathe, The Finite Element Analysis of Shells—Fundamentals. Computational Fluid and Solid Mechanics (Springer, Berlin, 2011)
M.A. Fernández, Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit. S \({\boldsymbol {\mathrm{e}}}\) MA J. 55(55), 59–108 (2011)
M.A. Fernández, Incremental displacement-correction schemes for incompressible fluid-structure interaction: stability and convergence analysis. Numer. Math. 123(1), 21–65 (2013)
M.A. Fernández, J. Mullaert, Convergence analysis for a class of explicit coupling schemes in incompressible fluid-structure interaction (2014). Research report RR-8670, Inria, 2015
M.A. Fernández, J. Mullaert, M. Vidrascu, Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures. Comput. Methods Appl. Mech. Eng. 267, 566–593 (2013)
M.A. Fernández, J. Mullaert, M. Vidrascu, Generalized Robin-Neumann explicit coupling schemes for incompressible fluid-structure interaction: stability analysis and numerics. Int. J. Numer. Methods Eng. 101(3), 199–229 (2015). doi:10.1002/nme.4785. https://hal.inria.fr/hal-00875819
C. Förster, W.A. Wall, E. Ramm, Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 196(7), 1278–1293 (2007)
G. Guidoboni, R. Glowinski, N. Cavallini, S. Čanić, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow. J. Comput. Phys. 228(18), 6916–6937 (2009)
M. Heil, A.L. Hazel, Fluid-structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162 (2011)
P. Kalita, R. Schaefer. Mechanical models of artery walls. Arch. Comput. Methods Eng.15(1), 1–36 (2008)
U. Küttler, C. Förster, W.A. Wall, A solution for the incompressibility dilemma in partitioned fluid-structure interaction with pure Dirichlet fluid domains. Comput. Mech. 38, 417–429 (2006)
M. Lombardi, N. Parolini, A. Quarteroni, G. Rozza, Numerical simulation of sailing boats: dynamics, FSI, and shape optimization, in Variational Analysis and Aerospace Engineering: Mathematical Challenges for Aerospace Design, ed. by G. Buttazzo, A. Frediani. Springer Optimization and Its Applications (Springer, Berlin, 2012), pp. 339–377
M. Lukáčová-Medvid’ová, G. Rusnáková, A. Hundertmark-Zaušková, Kinematic splitting algorithm for fluid-structure interaction in hemodynamics. Comput. Methods Appl. Mech. Eng. 265(1), 83–106 (2013)
P. Moireau, N. Xiao, M. Astorino, C.A. Figueroa, D. Chapelle, C.A. Taylor, J.-F. Gerbeau, External tissue support and fluid-structure simulation in blood flows. Biomech. Model. Mechanobiol. 11, 1–18 (2012)
M.P. Païdoussis, S.J. Price, E. de Langre, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities (Cambridge University Press, Cambridge, 2011)
K. Takizawa, T.E. Tezduyar, Computational methods for parachute fluid-structure interactions. Arch. Comput. Methods Eng. 19, 125–169 (2012)
D. Valdez-Jasso, H.T. Banks, M.A. Haider, D. Bia, Y. Zocalo, R.L. Armentano, M.S. Olufsen, Viscoelastic models for passive arterial wall dynamics. Adv. Appl. Math. Mech. 1(2), 151–165 (2009)
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This work was supported by the French National Research Agency (ANR) through the EXIFSI project (ANR-12-JS01-0004)
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Fernández, M.A., Landajuela, M., Mullaert, J., Vidrascu, M. (2016). Robin-Neumann Schemes for Incompressible Fluid-Structure Interaction. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_6
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