Linearized hydro-elasticity: A numerical study

Lorena Bociu; Steven  Derochers; Daniel  Toundykov

doi:10.3934/eect.2016018

Article Contents

2016, Volume 5, Issue 4: 533-559. Doi: 10.3934/eect.2016018

This issue Previous Article A uniform discrete inf-sup inequality for finite element hydro-elastic models Next Article Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions

Linearized hydro-elasticity: A numerical study

1.
Department of Mathematics, NC State University, Raleigh, NC 27695
2.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695
3.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received: July 31, 2016

Revised: August 31, 2016

Published: November 2016

Abstract Related Papers Cited by

Abstract

In view of control and stability theory, a recently obtained linearization around a steady state of a fluid-structure interaction is considered. The linearization was performed with respect to an external forcing term and was derived in an earlier paper via shape optimization techniques. In contrast to other approaches, like transporting to a fixed reference configuration, or using transpiration techniques, the shape optimization route is most suited to incorporating the geometry of the problem into the analysis. This refined description brings up new terms---missing in the classical coupling of linear Stokes flow and linear elasticity---in the matching of the normal stresses and the velocities on the interface. Later, it was demonstrated that this linear PDE system generates a $C_0$ semigroup, however, unlike in the standard Stokes-elasticity coupling, the wellposedness result depended on the fluid's viscosity and the new boundary terms which, among other things, involve the curvature of the interface. Here, we implement a finite element scheme for approximating solutions of this fluid-elasticity dynamics and numerically investigate the dependence of the discretized model on the ``new" terms present therein, in contrast with the classical Stokes-linear elasticity system.

Keywords:

Mathematics Subject Classification: Primary: 74F10, 65L60; Secondary: 93B18, 65L10, 35R35, 35R37, 74B20, 35Q30.

Citation:

Related Papers

Cited by

References

[1]	G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method, Appl. Math. (Warsaw), 35 (2008), 259-280.doi: 10.4064/am35-3-2.
[2]	G. Avalos and D. Toundykov, A uniform discrete inf-sup inequality for finite element hydro-elastic models, 2016, To appear in Evol. Equ. Control Theory, 2016.
[3]	H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, Journal of Mathematical Fluid Mechanics, 6 (2004), 21-52.doi: 10.1007/s00021-003-0082-5.
[4]	M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224.doi: 10.1007/BF01399555.
[5]	L. Bociu, D. Toundykov and J.-P. Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction, SIAM J. Math. Anal., 47 (2015), 1958-2000.doi: 10.1137/140970689.
[6]	L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, in Modern aspects of the theory of partial differential equations, vol. 216 of Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 2011, 93-120.doi: 10.1007/978-3-0348-0069-3_6.
[7]	L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction, Evol. Equ. Control Theory, 2 (2013), 55-79, To appear in Evolution Equations and Control Theory.doi: 10.3934/eect.2013.2.55.
[8]	M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., 9 (2007), 262-294.doi: 10.1007/s00021-005-0201-7.
[9]	J. Chazarain and A. Piriou, Caractérisation des problèmes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237.doi: 10.5802/aif.438.
[10]	P. G. Ciarlet, Mathematical Elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988, Three-dimensional elasticity.
[11]	C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires, Editions Eyrolles, Paris, 1994.
[12]	C. Conca, J. San Martín H. and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042.doi: 10.1080/03605300008821540.
[13]	D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.doi: 10.1007/s00205-005-0385-2.
[14]	B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71.doi: 10.1007/s002050050136.
[15]	B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538.doi: 10.5209/rev_REMA.2001.v14.n2.17030.
[16]	J. Donea, S. Giuliani and J. Halleux, An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering, 33 (1982), 689-723.doi: 10.1016/0045-7825(82)90128-1.
[17]	T. Fanion, M. Fernández and P. le Tallec, Deriving adequate formulations for fluid-structure interaction problems: From ALE to transpiration, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 51-79.
[18]	E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3 (2003), 419-441, Dedicated to Philippe Bénilan.doi: 10.1007/s00028-003-0110-1.
[19]	M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. I. Formulation and mathematical analysis, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4805-4835.doi: 10.1016/j.cma.2003.07.001.
[20]	M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. II. Numerical analysis and applications, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4837-4873.doi: 10.1016/j.cma.2003.08.001.
[21]	M. A. Fernández and M. Moubachir, An exact block-Newton algorithm for solving fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 336 (2003), 681-686.doi: 10.1016/S1631-073X(03)00151-1.
[22]	A. R. Galper and T. Miloh, Motion stability of a deformable body in an ideal fluid with applications to the N spheres problem, Phys. Fluids, 10 (1998), 119-130.doi: 10.1063/1.869570.
[23]	A. Georgescu, Hydrodynamic Stability Theory, vol. 9 of Mechanics: Analysis, Martinus Nijhoff Publishers, Dordrecht, 1985, Translated from the Romanian, Translation edited by David Sattinger.doi: 10.1007/978-94-017-1814-1.
[24]	C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid Mech., 4 (2002), 76-94.doi: 10.1007/s00021-002-8536-9.
[25]	C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636.doi: 10.1051/m2an:2000159.
[26]	C. Grandmont and Y. Maday, Fluid-structure interaction: A theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 1-22.
[27]	M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266.doi: 10.1007/PL00000954.
[28]	K. C. Hall and W. S. Clark, Linearized euler predictions of unsteady aerodynamic loads in cascades, AIAA Journal, 31 (1993), 540-550.doi: 10.2514/3.11363.
[29]	K. C. Hall and E. F. Crawley, Calculation of unsteady flows in turbomachinery using the linearizedeuler equations, AIAA Journal, 27 (1989), 777-787.doi: 10.2514/3.10178.
[30]	K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648.
[31]	T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29 (1981), 329-349.doi: 10.1016/0045-7825(81)90049-9.
[32]	M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499.doi: 10.1088/0951-7715/27/3/467.
[33]	M. Ikawa, A mixed problem for hyperbolic equations of second order with non-homogeneous Neumann type boundary condition, Osaka J. Math., 6 (1969), 339-374.
[34]	M. Lesoinne and C. Farhat, Re-engineering of an aeroelastic code for solving eigen problems in all flight regimes, in 4th European Computational Fluid Dynamics Conference, Athens, Greece, 1998, 1052-1061.
[35]	M. Lesoinne, M. Sarkis, U. Hetmaniuk and C. Farhat, A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3121-3146, Advances in Computational Methods for Fluid-Structure Interaction.doi: 10.1016/S0045-7825(00)00385-6.
[36]	M. J. Lighthill, On displacement thickness, J. Fluid Mech., 4 (1958), 383-392.doi: 10.1017/S0022112058000525.
[37]	G. Mortchéléwicz, Application of linearized euler equations to flutter, in 85th AGARD SMP Meeting, Aalborg, Denmark, 1997.
[38]	B. Palmerio, A two-dimensional FEM adaptive moving-node method for steady Euler flow simulations, Computer Methods in Applied Mechanics and Engineering, 71 (1988), 315-340.doi: 10.1016/0045-7825(88)90038-2.
[39]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, 1983.doi: 10.1007/978-1-4612-5561-1.
[40]	S. Piperno and C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 23-49.
[41]	P. Raj and B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis, in AIAA 24th Applied Aerodynamics Conference, 93-3506, Monterey, Canada, 1993.
[42]	R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge, 1982, Translated from the Japanese by Katsumi Miyahara.
[43]	J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147.doi: 10.1007/s002050100172.
[44]	T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77.doi: 10.1007/s00021-003-0083-4.
[45]	N. Takashi and T. J. Hughes, An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Computer Methods in Applied Mechanics and Engineering, 95 (1992), 115-138.doi: 10.1016/0045-7825(92)90085-X.
[46]	P. L. Tallec and J. Mouro, Fluid structure interaction with large structural displacements, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3039-3067, Advances in Computational Methods for Fluid-Structure Interaction.doi: 10.1016/S0045-7825(00)00381-9.
[47]	J. Tambača, M. Kosor, S. Čanić and D. Paniagua, Mathematical modeling of vascular stents, SIAM J. Appl. Math., 70 (2010), 1922-1952.doi: 10.1137/080722618.
[48]	C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. & Fluids, 1 (1973), 73-100.
[49]	R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO Anal. Numér., 18 (1984), 175-182.
[50]	T. Wick and W. Wollner, On the Differentiability of Fluid-Structure Interaction Problems with Respect to the Problem Data, Technical Report 2014-16, RICAM, 2014.