Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

doi:10.1007/s42967-019-0004-0

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (1): 61-80.doi: 10.1007/s42967-019-0004-0

• ORIGINAL PAPERS • 上一篇下一篇

Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

Ying He, Jie Shen

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

收稿日期:2018-06-08 修回日期:2018-11-16 出版日期:2019-03-20 发布日期:2019-05-11
通讯作者: Jie Shen E-mail:shen7@purdue.edu
基金资助:
This work is partially supported by NSF DMS-1620262,DMS-1720442 and AFOSR FA9550-16-1-0102.

Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

Ying He, Jie Shen

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received:2018-06-08 Revised:2018-11-16 Online:2019-03-20 Published:2019-05-11
Contact: Jie Shen E-mail:shen7@purdue.edu
Supported by:
This work is partially supported by NSF DMS-1620262,DMS-1720442 and AFOSR FA9550-16-1-0102.

摘要/Abstract

摘要： We consider in this paper numerical approximation of a nonlinear fluid-structure interaction (FSI) model with a fixed interface. We construct a new class of pressure-correction schemes for the FSI problem, 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.

关键词: Fluid-structure interaction, Pressure correction, Stability analysis

Abstract: We consider in this paper numerical approximation of a nonlinear fluid-structure interaction (FSI) model with a fixed interface. We construct a new class of pressure-correction schemes for the FSI problem, 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.

Key words: Fluid-structure interaction, Pressure correction, Stability analysis

中图分类号:

Ying He, Jie Shen. Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 61-80.

参考文献

1. Badia, S., Codina, R.:On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Int. J. Numer. Methods Eng. 72(1), 46-71(2007)
2. Badia, S., Nobile, F., Vergara, C.:Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227(14), 7027-7051(2008)
3. Badia, S., Quaini, A., Quarteroni, A.:Splitting methods based on algebraic factorization for fluidstructure interaction. SIAM J. Sci. Comput. 30(4), 1778-1805(2008)
4. Burman, E., 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)
5. Causin, P., Gerbeau, J.F., Nobile, F.:Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194(42-44), 4506-4527(2005)
6. Chakrabarti, S.K., Hernandez, S., Brebbia, C.A.: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)
7. Chorin, A.J.:Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745-762(1968)
8. Dong, S., Shen, J.:An unconditionally stable rotational velocity-correction scheme for incompressible flows. J. Comput. Phys. 229(19), 7013-7029(2010)
9. Farhat, C., Lesoinne, M., 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. Eng. 157(1-2), 95-114(1998)
10. Felippa, C.A., Park, K., Farhat, C.:Partitioned analysis of coupled mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 3247-3270(2001)
11. Fernández, M.A.:Coupling schemes for incompressible fluid-structure interaction:implicit, semiimplicit and explicit. SeMA J 55, 59-108(2011)
12. Fernández, M.A., Gerbeau, J.-F., Grandmont, C.:A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69(4), 794-8212007)
13. Guermond, J.L., Minev, P., Shen, J.:Error analysis of pressure-correction schemes for the timedependent Stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239-258(2005). (electronic)
14. Guermond, J.L., Minev, P., Shen, J.:An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44-47), 6011-6045(2006)
15. Guermond, J.L., Shen, J.:On the error estimates for the rotational pressure-correction projection methods. Math. Comput. 73(248), 1719-1737(2004). (electronic)
16. He, Y., Nicholls, D.P., Shen, J.:An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure. J. Comput. Phys. 231(8), 3007-3022(2012)
17. He, Y., Shen, J.:Unconditionally stable pressure-correction schemes for a linear fluid-structure interaction problem. Numer. Math. Theory Methods Appl. 7(4), 537-554(2014)
18. Hou, G., Wang, J., Layton, A.:Numerical methods for fluid-structure interaction-a review. Commun. Comput. Phys. 12(2), 337-377(2012)
19. Hron, J., Turek, S.:A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics. Springer, Berlin, Heidelberg (2006)
20. Hübner, B., Walhorn, E., Dinkler, D.:A monolithic approach to fluid-structure interaction using spacetime finite elements. Comput. Methods Appl. Mech. Eng. 193(23), 2087-2104(2004)
21. Ignatova, M., Kukavica, I., Lasiecka, I., Tuffaha, A.:On well-posedness for a free boundary fluidstructure model. J. Math. Phys. 53(11), 115624(2012)
22. Kukavica, I., Tuffaha, A., Ziane, M.:Strong solutions to a nonlinear fluid structure interaction system. J. Differ. Equ. 247(5), 1452-1478(2009)
23. Küttler, U., Wall, W.A.:Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput. Mech. 43(1), 61-72(2008)
24. Matthies, H.G., Steindorf, J.:Partitioned strong coupling algorithms for fluid-structure interaction. Comput. Struct. 81, 805-812(2003)
25. Pyo, J.-H.:Error estimates for the second order semi-discrete stabilized Gauge-Uzawa method for the Navier-Stokes equations. Int. J. Numer. Anal. Model. 10(1), 24-41(2013)
26. Quarteroni, A., Valli, A.:Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York (1999)
27. 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)
28. Temam, R.:Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires i. Arch. Rat. Mech. Anal. 32, 135-153(1969)
29. Toselli, A., Widlund, O.:Domain decomposition methods-algorithms and theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

编辑推荐

Metrics

本文评价