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Unconditionally Stable Pressure-Correction Schemes for a Linear Fluid-Structure Interaction Problem

Published online by Cambridge University Press:  09 August 2018

Ying He  and
Jie Shen
Ying He*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
Jie Shen*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
*
Corresponding author. Email address: he14@math.purdue.edu
*Email:shen@math.purdue.edu
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Abstract

We consider in this paper numerical approximation of the linear Fluid-Structure Interaction (FSI). We construct a new class of pressure-correction schemes for the linear FSI problem with a fixed interface, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.

Keywords

74F1076D0565M1235Q30Fluid-structure interactionpressure-correctionstability analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 4 , November 2014 , pp. 537 - 554
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Badia, S. and Codina, R.. On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Internat. J. Numer. Methods Engrg., 72(1):46–71, 2007.Google Scholar
[2] Badia, S., Nobile, F., and Vergara, C.. Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys., 227(14):7027–7051, 2008.Google Scholar
[3] Badia, S., Quaini, A., and Quarteroni, A.. Splitting methods based on algebraic factorization for fluid-structure interaction. SIAMJ. Sci. Comput., 30(4):1778–1805, 2008.Google Scholar
[4] Burman, E. and Fernández, M. A.. Stabilized explicit coupling for fluid-structure interaction using Nitsche's method. C. R. Math. Acad. Sci. Paris, 345(8):467–472, 2007.Google Scholar
[5] Chakrabarti, S. K., Hernandez, S., and Brebbia, C. A., editors. Fluid structure interaction and moving boundary problems, volume 43 of Advances in Fluid Mechanics. WIT Press, Southampton, 2005. Edited papers from the 3rd International Conference on Fluid Structure Interaction and the 8th International Conference on Computational Modelling and Experimental Measurements of Free and Moving Boundary Problems held in La Coruna, September 19-21, 2005.Google Scholar
[6] Du, Q., Gunzburger, M. D., Hou, L. S., and Lee, J.. Semidiscrete finite element approximations of a linear fluid-structure interaction problem. SIAM J. Numer. Anal, 42(1):1–29, 2004.Google Scholar
[7] Farhat, C., Lesoinne, M., and LeTallec, P.. Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput. Methods Appl. Mech. Engrg., 157(1-2) :95–114, 1998.Google Scholar
[8] Felippa, C. A., Park, K., and Farhat, C.. Partitioned analysis of coupled mechanical systems. 190:32473270, 2001.Google Scholar
[9] Fernández, M. A.. Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit. S MAJ., (55):59–108, 2011.Google Scholar
[10] Fernández, M. A., Gerbeau, J.-F., and Grandmont, C.. A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Internat. J. Numer. Methods Engrg., 69(4):794–821, 2007.Google Scholar
[11] Guermond, J. L., Minev, P., and Shen, J.. Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions. SIAM J. Numer. Anal, 43(1):239–258 (electronic), 2005.Google Scholar
[12] Guermond, J. L., Minev, P., and Shen, J.. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg., 195(44-47):6011–6045, 2006.Google Scholar
[13] He, Y., Nicholls, D. P., and Shen, J.. An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure. J. Comput. Phys., 231(8):3007–3022, 2012.CrossRefGoogle Scholar
[14] Hou, G., Wang, J., and Layton, A.. Numerical methods for fluid-structure interaction—a review. Commun. Comput. Phys., 12(2):337–377, 2012.CrossRefGoogle Scholar
[15] Hron, J. and Turek, S.. A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics. Springer, 2006.Google Scholar
[16] Hübner, B., Walhorn, E., and Dinkler, D.. A monolithic approach to fluid-structure interaction using space-time finite elements. Computer methods in applied mechanics and engineering, 193(23):2087–2104, 2004.CrossRefGoogle Scholar
[17] Kukavica, I., Tuffaha, A., and Ziane, M.. Strong solutions to a nonlinear fluid structure interaction system. J. Differential Equations, 247(5):1452–1478, 2009.Google Scholar
[18] Küttler, U. and Wall, W. A.. Fixed-point fluid–structure interaction solvers with dynamic relaxation. Computational Mechanics, 43(1):61–72, 2008.Google Scholar
[19] Matthies, H. G. and Steindorf, J.. Partitioned strong coupling algorithms for fluidstructure interaction. 81:805–812, 2003.Google Scholar
[20] Quarteroni, A. and Valli, A.. Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 1999. Oxford Science Publications.Google Scholar
[21] Shen, J.. Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput., 15(6):1489–1505, 1994.Google Scholar
[22] Toselli, A. and Widlund, O.. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.Google Scholar