December  2016, 5(4): 533-559. doi: 10.3934/eect.2016018

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, United States

3. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States

Received  August 2016 Revised  September 2016 Published  October 2016

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.
Citation: Lorena Bociu, Steven Derochers, Daniel Toundykov. Linearized hydro-elasticity: A numerical study. Evolution Equations & Control Theory, 2016, 5 (4) : 533-559. doi: 10.3934/eect.2016018
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.  doi: 10.4064/am35-3-2.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00021-003-0082-5.  Google Scholar

[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.  doi: 10.1007/BF01399555.  Google Scholar

[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.  doi: 10.1137/140970689.  Google Scholar

[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, (2011), 93.  doi: 10.1007/978-3-0348-0069-3_6.  Google Scholar

[7]

L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction,, Evol. Equ. Control Theory, 2 (2013), 55.  doi: 10.3934/eect.2013.2.55.  Google Scholar

[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.  doi: 10.1007/s00021-005-0201-7.  Google Scholar

[9]

J. Chazarain and A. Piriou, Caractérisation des problèmes mixtes hyperboliques bien posés,, Ann. Inst. Fourier (Grenoble), 22 (1972), 193.  doi: 10.5802/aif.438.  Google Scholar

[10]

P. G. Ciarlet, Mathematical Elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1988).   Google Scholar

[11]

C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires,, Editions Eyrolles, (1994).   Google Scholar

[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.  doi: 10.1080/03605300008821540.  Google Scholar

[13]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[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.  doi: 10.1007/s002050050136.  Google Scholar

[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.  doi: 10.5209/rev_REMA.2001.v14.n2.17030.  Google Scholar

[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.  doi: 10.1016/0045-7825(82)90128-1.  Google Scholar

[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, (2003), 51.   Google Scholar

[18]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419.  doi: 10.1007/s00028-003-0110-1.  Google Scholar

[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.  doi: 10.1016/j.cma.2003.07.001.  Google Scholar

[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.  doi: 10.1016/j.cma.2003.08.001.  Google Scholar

[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.  doi: 10.1016/S1631-073X(03)00151-1.  Google Scholar

[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.  doi: 10.1063/1.869570.  Google Scholar

[23]

A. Georgescu, Hydrodynamic Stability Theory, vol. 9 of Mechanics: Analysis,, Martinus Nijhoff Publishers, (1985).  doi: 10.1007/978-94-017-1814-1.  Google Scholar

[24]

C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid Mech., 4 (2002), 76.  doi: 10.1007/s00021-002-8536-9.  Google Scholar

[25]

C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609.  doi: 10.1051/m2an:2000159.  Google Scholar

[26]

C. Grandmont and Y. Maday, Fluid-structure interaction: A theoretical point of view,, in Fluid-structure interaction, (2003), 1.   Google Scholar

[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.  doi: 10.1007/PL00000954.  Google Scholar

[28]

K. C. Hall and W. S. Clark, Linearized euler predictions of unsteady aerodynamic loads in cascades,, AIAA Journal, 31 (1993), 540.  doi: 10.2514/3.11363.  Google Scholar

[29]

K. C. Hall and E. F. Crawley, Calculation of unsteady flows in turbomachinery using the linearizedeuler equations,, AIAA Journal, 27 (1989), 777.  doi: 10.2514/3.10178.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0045-7825(81)90049-9.  Google Scholar

[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.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[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.   Google Scholar

[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, (1998), 1052.   Google Scholar

[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.  doi: 10.1016/S0045-7825(00)00385-6.  Google Scholar

[36]

M. J. Lighthill, On displacement thickness,, J. Fluid Mech., 4 (1958), 383.  doi: 10.1017/S0022112058000525.  Google Scholar

[37]

G. Mortchéléwicz, Application of linearized euler equations to flutter,, in 85th AGARD SMP Meeting, (1997).   Google Scholar

[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.  doi: 10.1016/0045-7825(88)90038-2.  Google Scholar

[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.  Google Scholar

[40]

S. Piperno and C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems,, in Fluid-structure interaction, (2003), 23.   Google Scholar

[41]

P. Raj and B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis,, in AIAA 24th Applied Aerodynamics Conference, (1993), 93.   Google Scholar

[42]

R. Sakamoto, Hyperbolic Boundary Value Problems,, Cambridge University Press, (1982).   Google Scholar

[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.  doi: 10.1007/s002050100172.  Google Scholar

[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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar

[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.  doi: 10.1016/0045-7825(92)90085-X.  Google Scholar

[46]

P. L. Tallec and J. Mouro, Fluid structure interaction with large structural displacements,, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3039.  doi: 10.1016/S0045-7825(00)00381-9.  Google Scholar

[47]

J. Tambača, M. Kosor, S. Čanić and D. Paniagua, Mathematical modeling of vascular stents,, SIAM J. Appl. Math., 70 (2010), 1922.  doi: 10.1137/080722618.  Google Scholar

[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.   Google Scholar

[49]

R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations,, RAIRO Anal. Numér., 18 (1984), 175.   Google Scholar

[50]

T. Wick and W. Wollner, On the Differentiability of Fluid-Structure Interaction Problems with Respect to the Problem Data,, Technical Report 2014-16, (2014), 2014.   Google Scholar

show all references

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.  doi: 10.4064/am35-3-2.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00021-003-0082-5.  Google Scholar

[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.  doi: 10.1007/BF01399555.  Google Scholar

[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.  doi: 10.1137/140970689.  Google Scholar

[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, (2011), 93.  doi: 10.1007/978-3-0348-0069-3_6.  Google Scholar

[7]

L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction,, Evol. Equ. Control Theory, 2 (2013), 55.  doi: 10.3934/eect.2013.2.55.  Google Scholar

[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.  doi: 10.1007/s00021-005-0201-7.  Google Scholar

[9]

J. Chazarain and A. Piriou, Caractérisation des problèmes mixtes hyperboliques bien posés,, Ann. Inst. Fourier (Grenoble), 22 (1972), 193.  doi: 10.5802/aif.438.  Google Scholar

[10]

P. G. Ciarlet, Mathematical Elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1988).   Google Scholar

[11]

C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires,, Editions Eyrolles, (1994).   Google Scholar

[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.  doi: 10.1080/03605300008821540.  Google Scholar

[13]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[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.  doi: 10.1007/s002050050136.  Google Scholar

[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.  doi: 10.5209/rev_REMA.2001.v14.n2.17030.  Google Scholar

[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.  doi: 10.1016/0045-7825(82)90128-1.  Google Scholar

[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, (2003), 51.   Google Scholar

[18]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419.  doi: 10.1007/s00028-003-0110-1.  Google Scholar

[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.  doi: 10.1016/j.cma.2003.07.001.  Google Scholar

[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.  doi: 10.1016/j.cma.2003.08.001.  Google Scholar

[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.  doi: 10.1016/S1631-073X(03)00151-1.  Google Scholar

[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.  doi: 10.1063/1.869570.  Google Scholar

[23]

A. Georgescu, Hydrodynamic Stability Theory, vol. 9 of Mechanics: Analysis,, Martinus Nijhoff Publishers, (1985).  doi: 10.1007/978-94-017-1814-1.  Google Scholar

[24]

C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid Mech., 4 (2002), 76.  doi: 10.1007/s00021-002-8536-9.  Google Scholar

[25]

C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609.  doi: 10.1051/m2an:2000159.  Google Scholar

[26]

C. Grandmont and Y. Maday, Fluid-structure interaction: A theoretical point of view,, in Fluid-structure interaction, (2003), 1.   Google Scholar

[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.  doi: 10.1007/PL00000954.  Google Scholar

[28]

K. C. Hall and W. S. Clark, Linearized euler predictions of unsteady aerodynamic loads in cascades,, AIAA Journal, 31 (1993), 540.  doi: 10.2514/3.11363.  Google Scholar

[29]

K. C. Hall and E. F. Crawley, Calculation of unsteady flows in turbomachinery using the linearizedeuler equations,, AIAA Journal, 27 (1989), 777.  doi: 10.2514/3.10178.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0045-7825(81)90049-9.  Google Scholar

[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.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[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.   Google Scholar

[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, (1998), 1052.   Google Scholar

[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.  doi: 10.1016/S0045-7825(00)00385-6.  Google Scholar

[36]

M. J. Lighthill, On displacement thickness,, J. Fluid Mech., 4 (1958), 383.  doi: 10.1017/S0022112058000525.  Google Scholar

[37]

G. Mortchéléwicz, Application of linearized euler equations to flutter,, in 85th AGARD SMP Meeting, (1997).   Google Scholar

[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.  doi: 10.1016/0045-7825(88)90038-2.  Google Scholar

[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.  Google Scholar

[40]

S. Piperno and C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems,, in Fluid-structure interaction, (2003), 23.   Google Scholar

[41]

P. Raj and B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis,, in AIAA 24th Applied Aerodynamics Conference, (1993), 93.   Google Scholar

[42]

R. Sakamoto, Hyperbolic Boundary Value Problems,, Cambridge University Press, (1982).   Google Scholar

[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.  doi: 10.1007/s002050100172.  Google Scholar

[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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar

[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.  doi: 10.1016/0045-7825(92)90085-X.  Google Scholar

[46]

P. L. Tallec and J. Mouro, Fluid structure interaction with large structural displacements,, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3039.  doi: 10.1016/S0045-7825(00)00381-9.  Google Scholar

[47]

J. Tambača, M. Kosor, S. Čanić and D. Paniagua, Mathematical modeling of vascular stents,, SIAM J. Appl. Math., 70 (2010), 1922.  doi: 10.1137/080722618.  Google Scholar

[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.   Google Scholar

[49]

R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations,, RAIRO Anal. Numér., 18 (1984), 175.   Google Scholar

[50]

T. Wick and W. Wollner, On the Differentiability of Fluid-Structure Interaction Problems with Respect to the Problem Data,, Technical Report 2014-16, (2014), 2014.   Google Scholar

[1]

Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55

[2]

Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122

[3]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

[4]

Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181

[5]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89

[6]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184

[7]

Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure & Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95

[8]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013

[9]

George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817

[10]

Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102

[11]

Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086

[12]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087

[13]

Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks & Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397

[14]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39

[15]

Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269

[16]

Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525

[17]

Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks & Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109

[18]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633

[19]

Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems & Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057

[20]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

[Back to Top]