# American Institute of Mathematical Sciences

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 and Control Theory, 2016, 5 (4) : 533-559. doi: 10.3934/eect.2016018
##### References:

show all references

##### References:
 [1] Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817 [10] Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 [11] Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102 [12] 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 and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086 [13] Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525 [14] Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks and Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397 [15] 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 [16] Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269 [17] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [18] Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633 [19] Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109 [20] Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems and Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057

2021 Impact Factor: 1.169