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A weak-coupling immersed boundary method for fluid-structure interaction with low density ratio of solid to fluid. (English) Zbl 1383.74086

Summary: We present a weak-coupling approach for fluid-structure interaction with low density ratio (\(\rho\)) of solid to fluid. For accurate and stable solutions, we introduce predictors, an explicit two-step method and the implicit Euler method, to obtain provisional velocity and position of fluid-structure interface at each time step, respectively. The incompressible Navier-Stokes equations, together with these provisional velocity and position at the fluid-structure interface, are solved in an Eulerian coordinate using an immersed-boundary finite-volume method on a staggered mesh. The dynamic equation of an elastic solid-body motion, together with the hydrodynamic force at the provisional position of the interface, is solved in a Lagrangian coordinate using a finite element method. Each governing equation for fluid and structure is implicitly solved using second-order time integrators. The overall second-order temporal accuracy is preserved even with the use of lower-order predictors. A linear stability analysis is also conducted for an ideal case to find the optimal explicit two-step method that provides stable solutions down to the lowest density ratio. With the present weak coupling, three different fluid-structure interaction problems were simulated: flows around an elastically mounted rigid circular cylinder, an elastic beam attached to the base of a stationary circular cylinder, and a flexible plate, respectively. The lowest density ratios providing stable solutions are searched for the first two problems and they are much lower than 1 (\(\rho_{\min} = 0.21\) and 0.31, respectively). The simulation results agree well with those from strong coupling suggested here and also from previous numerical and experimental studies, indicating the efficiency and accuracy of the present weak coupling.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Hou, G.; Wang, J.; Layton, A., Numerical methods for fluid-structure interaction - a review, Commun. Comput. Phys., 12, 2, 337-377 (2012) · Zbl 1373.76001
[2] Felippa, C. A.; Park, K.; Farhat, C., Partitioned analysis of coupled mechanical systems, Comput. Methods Appl. Mech. Eng., 190, 3247-3270 (2001) · Zbl 0985.76075
[3] Weeratunga, S.; Pramono, E., Direct Coupled Aeroelastic Analysis Through Concurrent Implicit Time Integration on a Parallel Computer (April 1994), AIAA Paper No. 94-1550
[4] Borazjani, I.; Ge, L.; Sotiropoulos, F., Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies, J. Comput. Phys., 227, 7587-7620 (2008) · Zbl 1213.76129
[5] Le Tallec, P.; Mouro, J., Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Eng., 190, 3039-3067 (2001) · Zbl 1001.74040
[6] Deparis, S.; Fernández, M. A.; Formaggia, L., Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions, Math. Model. Numer. Anal., 37, 601-616 (2003) · Zbl 1118.74315
[7] Küttler, U.; Wall, W. A., Fixed-point fluid-structure interaction solvers with dynamic relaxation, Comput. Mech., 43, 61-72 (2008) · Zbl 1236.74284
[8] Kassiotis, C.; Ibrahimbegovic, A.; Niekamp, R.; Matthies, H. G., Nonlinear fluid-structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability proof and validation examples, Comput. Mech., 47, 305-323 (2011) · Zbl 1398.74084
[9] Breuer, M.; De Nayer, G.; Münsch, M.; Gallinger, T.; Wüchner, R., Fluid-structure interaction using a partitioned semi-implicit predictor-corrector coupling scheme for the application of large-eddy simulation, J. Fluids Struct., 29, 107-130 (2012)
[10] Tian, F.-B.; Dai, H.; Luo, H.; Doyle, J. F.; Rousseau, B., Fluid-structure interaction involving large deformations: 3D simulations and applications to biological systems, J. Comput. Phys., 258, 451-469 (2014) · Zbl 1349.76274
[11] Gilmanov, A.; Le, T. B.; Sotiropoulos, F., A numerical approach for simulating fluid structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains, J. Comput. Phys., 300, 814-843 (2015) · Zbl 1349.74323
[12] Tezduyar, T. E., Finite element methods for flow problems with moving boundaries and interfaces, Arch. Comput. Methods Eng., 8, 83-130 (2001) · Zbl 1039.76037
[13] Gerbeau, J.; Vidrascu, M., A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows, Math. Model. Numer. Anal., 37, 631-647 (2003) · Zbl 1070.74047
[14] Matthies, H. G.; Steindorf, J., Partitioned strong coupling algorithms for fluid-structure interaction, Comput. Struct., 81, 805-812 (2003)
[15] Fernández, M.Á.; Moubachir, M., A Newton method using exact jacobians for solving fluid-structure coupling, Comput. Struct., 83, 127-142 (2005)
[16] Piperno, S.; Farhat, C., Partitioned procedures for the transient solution of coupled aeroelastic problems - part II: energy transfer analysis and three-dimensional applications, Comput. Methods Appl. Mech. Eng., 190, 3147-3170 (2001) · Zbl 1015.74009
[17] Piperno, S.; Farhat, C.; Larrouturou, B., Partitioned procedures for the transient solution of coupled aeroelastic problems - part I: model problem, theory and two-dimensional application, Comput. Methods Appl. Mech. Eng., 124, 79-112 (1995) · Zbl 1067.74521
[18] Farhat, C.; van der Zee, K. G.; Geuzaine, P., Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity, Comput. Methods Appl. Mech. Eng., 195, 1973-2001 (2006) · Zbl 1178.76259
[19] Farhat, C.; Rallu, A.; Wang, K.; Belytschko, T., Robust and provably second-order explicit-explicit and implicit-explicit staggered time-integrators for highly nonlinear compressible fluid-structure interaction problems, Int. J. Numer. Methods Eng., 84, 73-107 (2010) · Zbl 1202.74167
[20] Dettmer, W. G.; Perić, D., A new staggered scheme for fluid-structure interaction, Int. J. Numer. Methods Eng., 93, 1-22 (2013) · Zbl 1352.74471
[21] Yang, J.; Preidikman, S.; Balaras, E., A strongly coupled, embedded-boundary method for fluid-structure interactions of elastically mounted rigid bodies, J. Fluids Struct., 24, 167-182 (2008)
[22] Causin, P.; Gerbeau, J.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Eng., 194, 4506-4527 (2005) · Zbl 1101.74027
[23] Förster, C.; Wall, W. A.; Ramm, E., Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 196, 1278-1293 (2007) · Zbl 1173.74418
[24] van Brummelen, E. H., Added mass effects of compressible and incompressible flows in fluid-structure interaction, J. Appl. Mech., 76, 2, Article 021206 pp. (2009)
[25] Peskin, C. S., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 252-271 (1972) · Zbl 0244.92002
[26] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 239-261 (2005) · Zbl 1117.76049
[27] Beyer, R. P.; LeVeque, R. J., Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Numer. Anal., 29, 332-364 (1992) · Zbl 0762.65052
[28] Saiki, E. M.; Biringen, S., Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. Comput. Phys., 123, 2, 450-465 (1996) · Zbl 0848.76052
[29] Lai, M.-C.; Peskin, C. S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys., 160, 2, 705-719 (2000) · Zbl 0954.76066
[30] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip flow boundary with an external force field, J. Comput. Phys., 105, 354-366 (1993) · Zbl 0768.76049
[31] Huang, W.-X.; Sung, H. J., An immersed boundary method for fluid-flexible structure interaction, Comput. Methods Appl. Mech. Eng., 198, 2650-2661 (2009) · Zbl 1228.74105
[32] Fadlun, E. A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., 161, 35-60 (2000) · Zbl 0972.76073
[33] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171, 132-150 (2001) · Zbl 1057.76039
[34] Lee, J.; You, D., An implicit ghost-cell immersed boundary method for simulations of moving body problems with control of spurious force oscillations, J. Comput. Phys., 233, 295-314 (2013)
[35] Luo, H.; Mittal, R.; Zheng, X.; Bielamowicz, S. A.; Walsh, R. J.; Hahn, J. K., An immersed-boundary method for flow-structure interaction in biological systems with application to phonation, J. Comput. Phys., 227, 9303-9332 (2008) · Zbl 1148.74048
[36] Lee, I.; Choi, H., A discrete-forcing immersed boundary method for the fluid-structure interaction of an elastic slender body, J. Comput. Phys., 280, 529-546 (2015) · Zbl 1349.76231
[37] Yang, J.; Balaras, E., An embedded-boundary formation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys., 215, 12-40 (2006) · Zbl 1140.76355
[38] Lee, J.; Kim, J.; Choi, H.; Yang, K.-S., Sources of spurious force oscillations from an immersed boundary method for moving-body problems, J. Comput. Phys., 230, 2677-2695 (2011) · Zbl 1316.76075
[39] Irons, B. M.; Tuck, R. C., A version of the Aitken accelerator for computer iteration, Int. J. Numer. Methods Eng., 1, 275-277 (1969) · Zbl 0256.65021
[40] Choi, H.; Moin, P., Effects of the computational time step on numerical solutions of turbulent flow, J. Comput. Phys., 113, 1-4 (1994) · Zbl 0807.76051
[41] Kim, K.; Baek, S.-J.; Sung, H. J., An implicit velocity decoupling procedure for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 38, 125-138 (2002) · Zbl 1059.76046
[42] Hinton, E.; Rock, T.; Zienkiewicz, O. C., A note on mass lumping and related processes in the finite element method, Earthq. Eng. Struct. Dyn., 4, 245-249 (1976)
[43] Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(α\) method, J. Appl. Mech., 60, 371-375 (1993) · Zbl 0775.73337
[44] Matthies, H.; Strang, G., The solution of nonlinear finite element equations, Int. J. Numer. Methods Eng., 14, 1613-1626 (1979) · Zbl 0419.65070
[45] Blackburn, H. M.; Karniadakis, G. E., Two-and three-dimensional simulations of vortex-induced vibration of a circular cylinder, (Third International Offshore and Polar Engineering Conference (1993)), 715-720
[46] Yang, J.; Stern, F., A non-iterative direct forcing immersed boundary method for strongly-coupled fluid-solid interactions, J. Comput. Phys., 295, 779-804 (2015) · Zbl 1349.76556
[47] Turek, S.; Hron, J., Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, (Bungartz, H. J.; Schäfer, M., Fluid-Structure Interaction: Modelling, Simulation, Optimisation (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, Netherlands), 371-385 · Zbl 1323.76049
[48] Luhar, M.; Nepf, H. M., Flow-induced reconfiguration of buoyant and flexible aquatic vegetation, Limnol. Oceanogr., 56, 2003-2017 (2011)
[49] Jeong, J.; Hussain, F., On the identification of a vortex, J. Fluid Mech., 285, 69-94 (1995) · Zbl 0847.76007
[50] Bistritz, Y., Zero location with respect to the unit circle of discrete-time linear system polynomials, Proc. IEEE, 72, 1131-1142 (1984)
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