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Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. (English) Zbl 1173.74418
Summary: Within this paper the so-called artificial added mass effect is investigated which is responsible for devastating instabilities within sequentially staggered Fluid-structure Interaction (FSI) simulations where incompressible fluids are considered.
A discrete representation of the added mass operator $$\mathcal M_A$$ is given and ‘instability conditions’ are evaluated for different temporal discretisation schemes. It is proven that for every sequentially staggered scheme and given spatial discretisation of a problem, a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable. The analysis is quite general and does not depend upon the particular spatial discretisation schemes used. However here special attention is given to stabilised finite elements employed on the fluid partition. Numerical investigations further highlight the results.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74S20 Finite difference methods applied to problems in solid mechanics 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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##### References:
 [1] D. Mok, Partitionierte Lösungsansätze in der Strukturdynamik und der Fluid-Struktur-Interaktion, Ph.D. thesis, Institut für Baustatik, Universität Stuttgart, 2001. [2] D. Mok, W. Wall, Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures, in: W.A. Wall, K.-U. Bletzinger, K. Schweizerhof (Eds.), Proceedings of Trends in Computational Structural Mechanics, 2001, pp. 689-698. [3] W. Wall, D. Mok, E. Ramm, Partitioned analysis approach of the transient coupled response of viscous fluids and flexible structures, in: W. Wunderlich (Ed.), Solids, Structures and Coupled Problems in Engineering, Proceedings of the European Conference on Computational Mechanics ECCM ’99, Munich, 1999. [4] Causin, P.; Gerbeau, J.-F.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid – structure problems, Comput. methods appl. mech. engrg., 194, 4506-4527, (2005) · Zbl 1101.74027 [5] Park, K.; Felippa, C.; DeRuntz, J., Stabilization of staggered solution procedures for fluid – structure interaction analysis, Amd, 26, 95-124, (1977) · Zbl 0389.76002 [6] Tezduyar, T.; Sathe, S.; Keedy, R.; Stein, K., Space – time finite element techniques for computation of fluid – structure interactions, Comput. methods appl. mech. engrg., 195, 2002-2027, (2006) · Zbl 1118.74052 [7] Le Tallec, P.; Mouro, J., Fluid structure interaction with large structural displacements, Comput. methods appl. mech. engrg., 190, 3039-3067, (2001) · Zbl 1001.74040 [8] S. Piperno, C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems, Revue Européenne des Éléments Finis, “Fluid – structure Interaction”, vol. 9, 2000, pp. 655-680. · Zbl 1003.74081 [9] Piperno, S.; Farhat, C., Partitioned procedures for the transient solution of coupled aeroelastic problems – part II: energy transfer analysis and three-dimensional applications, Comput. methods appl. mech. engrg., 190, 3147-3170, (2001) · Zbl 1015.74009 [10] Chung, J.; Hulbert, G., A time integration algorithm for structural dynamics with improved numerical dissipation; the generalized-α method, J. appl. math., 60, 371-375, (1993) · Zbl 0775.73337 [11] Piperno, S., Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2d inviscid aeroelastic simulations, Int. J. numer. methods fluids, 25, 1207-1226, (1997) · Zbl 0910.76065 [12] Gunzburger, M., Finite element methods for viscous incompressible flows: A guide to theory, practice and algorithms, (1989), Academic Press, Inc. [13] Franca, L.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. methods appl. mech. engrg., 123, 299-308, (1995) · Zbl 1067.76567 [14] Franca, L.; Valentin, F., On an improved unusual stabilized finite element method for the advective – reactive – diffusive equation, Comput. methods appl. mech. engrg., 190, 1785-1800, (2000) · Zbl 0976.76038 [15] Hughes, T.; Franca, L., A new finite element formulation for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. methods appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067 [16] Hughes, T.; Franca, L.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska-Brezzi condition: a stable petrov – galerkin formulation of the Stokes problem accommodating equal-order interpolation, Comput. methods appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077 [17] Baiocchi, C.; Brezzi, F., Virtual bubbles and Galerkin-least-squares type of methods (ga.L.S.), Comput. methods appl. mech. engrg., 105, 125-141, (1993) · Zbl 0772.76033 [18] Franca, L.; Farhat, C.; Lesoinne, M.; Russo, A., Unusual stabilized finite element methods and residual free bubbles, Int. J. numer. methods fluids, 27, 159-168, (1998) · Zbl 0904.76045 [19] Franca, L.; Hughes, T., Two classes of mixed finite element methods, Comput. methods appl. mech. engrg., 69, 89-129, (1988) · Zbl 0629.73053 [20] Mittal, S.; Tezduyar, T., A finite element study of incompressible flows past oscillating cylinders and airfoils, Int. J. numer. methods fluids, 15, 1073-1118, (1992) [21] Behr, M.; Franca, L.; Tezduyar, T., Stabilised finite element methods for the velocity – pressure – stress formulation of incompressible flows, Comput. methods appl. mech. engrg., 104, 31-48, (1993) · Zbl 0771.76033 [22] W. Wall, Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen, Ph.D. thesis, Institut für Baustatik, Universität Stuttgart, 1999. [23] Franca, L.; Oliveira, S., Pressure bubbles stabilization features in the Stokes problem, Comput. methods appl. mech. engrg., 192, 1929-1937, (2003) · Zbl 1029.76032 [24] Barrenechea, G.; Valentin, F., An unusual stabilized finite element method for a generalized Stokes problem, Numer. math., 92, 652-677, (2002) · Zbl 1019.65087 [25] M.A. Fernández, J.-F. Gerbeau, C. Grandmont, A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Int. J. Numer. Methods Engrg., in press, doi:10.1002/nme.1792. · Zbl 1194.74393
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