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A lattice Boltzmann based implicit immersed boundary method for fluid-structure interaction. (English) Zbl 1189.76407
Summary: The immersed boundary method is a practical and effective method for fluid-structure interaction problems. It has been applied to a variety of problems. Most of the time-stepping schemes used in the method are explicit, which suffer a drawback in terms of stability and restriction on the time step. We propose a lattice Boltzmann based implicit immersed boundary method where the immersed boundary force is computed at the unknown configuration of the structure at each time step. The fully nonlinear algebraic system resulting from discretizations is solved by an Inexact Newton-Krylov method in a Jacobian-free manner. The test problem of a flexible filament in a flowing viscous fluid is considered. Numerical results show that the proposed implicit immersed boundary method is much more stable with larger time steps and significantly outperforms the explicit version in terms of computational cost.

76M28 Particle methods and lattice-gas methods
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] C.S. Peskin, Flow patterns around heart valves: A digital computer method for solving the equations of motion, PhD thesis, Albert Einstein Coll. Med., 1972
[2] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220-252, (1977) · Zbl 0403.76100
[3] Eggleton, C.D.; Popel, A.S., Large deformation of red blood cell ghosts in a simple shear flow, Phys. fluids, 10, 1834-1845, (1998)
[4] Fogelson, A.L., Continuum models of platelet aggregation: formulation and mechanical properties, SIAM J. appl. math., 52, 1089-1110, (1992) · Zbl 0756.92013
[5] Dillon, R.; Fauci, L.; Gaver, D., A microscale model of bacterial swimming, chemotaxis and substrate transport, J. theor. biol., 177, 325-340, (1995)
[6] Fauci, L.J.; McDonald, A., Sperm motility in the presence of boundaries, B. math. biol., 57, 679-699, (1994) · Zbl 0826.92017
[7] Dillon, R.; Fauci, L., An integrative model of internal axoneme mechanics and external fluid dynamics in ciliary beating, J. theor. biol., 207, 415-430, (2000)
[8] Miller, L.A.; Peskin, C.S., When vortices stick: an aerodynamic transition in tiny insect flight, J. exp. biol., 207, 3073-3088, (2004)
[9] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309
[10] Zhu, L.; Peskin, C.S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. comput. phys., 179, 452-468, (2002) · Zbl 1130.76406
[11] LeVeque, R.J.; Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. sci. comput., 18, 709-735, (1997) · Zbl 0879.76061
[12] Lee, L.; LeVeque, R.J., An immersed interface method for incompressible navier – stokes equations, SIAM J. sci. comput., 25, 832-856, (2003) · Zbl 1163.65322
[13] Glowinski, R.; Pan, T.-W.; Hesla, T.I.; Joseph, D.D.; Periaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys., 169, 363-427, (2001) · Zbl 1047.76097
[14] Pan, T.-W.; Glowinski, R., Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow, J. comput. phys., 181, 260-279, (2002) · Zbl 1178.76235
[15] Hao, J.; Pan, T.-W.; Glowinski, R.; Joseph, D.D., A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach, J. non-Newtonian fluid mech., 156, 95-111, (2009) · Zbl 1274.76344
[16] Hao, J.; Pan, T.-W.; Canic, S.; Glowinski, R.; Rosenstrauch, D., A fluid – cell interaction and adhesion algorithm for tissue-coating of cardiovascular implants, Multiscale model. simul., 7, 1669-1694, (2009) · Zbl 1181.92045
[17] Peskin, C.S.; McQueen, D.M., Computational biofluid dynamics, Contemp. math., 141, 161-186, (1993) · Zbl 0786.76108
[18] Peskin, C.S.; McQueen, D.M., A general method for the computer simulation of biological systems interacting with fluids, Sympos. soc. exp. biol., 49, 265-276, (1995)
[19] Peskin, C.S.; McQueen, D.M., Fluid dynamics of the heart and its valves, (), 309-337
[20] McCracken, M.F.; Peskin, C.S., A vortex method for blood flow through heart valves, J. comput. phys., 35, 183-205, (1980) · Zbl 0428.92010
[21] Peskin, C.S.; Printz, B.F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 33-46, (1993) · Zbl 0762.92011
[22] Rosar, M.E.; Peskin, C.S., Fluid flow in collapsible elastic tubes: A three-dimensional numerical model, New York J. math., 7, 281-302, (2001) · Zbl 1051.76016
[23] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 509-534, (1999) · Zbl 0953.76069
[24] Lai, M.C.; Peskin, C.S., An immersed boundary method with formal second order accuracy and reduced numerical viscosity, J. comput. phys., 160, 705-719, (2000) · Zbl 0954.76066
[25] D.M. McQueen, C.S. Peskin, L. Zhu, The immersed boundary method for incompressible fluid – structure interaction, in: Proceedings of the First M.I.T. Conference on Computational Fluid and Solid Mechanics, Boston, USA, 2001, pp. 26-30
[26] Griffith, B.E.; Peskin, C.S., On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficient smooth problems, J. comput. phys., 208, 75-105, (2005) · Zbl 1115.76386
[27] L. Zhu, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, PhD thesis, Courant Institute, New York University, 2001 · Zbl 1130.76406
[28] Kim, Y.; Peskin, C.S., Penalty immersed boundary method for an elastic boundary with mass, Phys. fluids, 19, 053103, (2007) · Zbl 1146.76441
[29] Atzberger, P.J.; Kramer, P.R.; Peskin, C.S., A stochastic immersed boundary method for fluid – structure dynamics at microscopic length scales, J. comput. phys., 224, 1255-1292, (2007) · Zbl 1124.74052
[30] Atzberger, P.J.; Kramer, P.R., Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations, Math. comput. simul., 79, 379-408, (2008) · Zbl 1159.65006
[31] Stockie, J.M.; Wetton, B.R., Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes, J. comput. phys., 154, 41-64, (1999) · Zbl 0953.76070
[32] Zhu, L.; Peskin, C.S., Interaction of two flexible filaments in a flowing soap film, Phys. fluids, 15, 1954-1960, (2003) · Zbl 1186.76611
[33] Zhu, L., Scaling laws for drag of a compliant body in an incompressible viscous flow, J. fluid mech., 607, 387-400, (2008) · Zbl 1146.76020
[34] Givelberg, E., Modeling elastic shells immersed in fluid, Comm. pure appl. math., 57, 283-309, (2004) · Zbl 1118.74014
[35] Tu, C.; Peskin, C.S., Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods, SIAM J. sci. stat. comput., 13, 1361-1376, (1992) · Zbl 0760.76067
[36] Mayo, A.A.; Peskin, C.S., An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, Contemp. math., 141, 261-277, (1993) · Zbl 0787.76055
[37] Fauci, L.J.; Fogelson, A.L., Truncated Newton methods and the modeling of complex elastic structures, Comm. pure appl. math., 46, 787-818, (1993) · Zbl 0741.76103
[38] Taira, K.; Colonius, T., The immersed boundary method: A projection approach, J. comput. phys., 225, 2118-2137, (2007) · Zbl 1343.76027
[39] Hou, T.Y.; Shi, Z., An efficient semi-implicit immersed boundary method for the navier – stokes equations, J. comput. phys., 227, 8968-8991, (2008) · Zbl 1161.76048
[40] Mori, Y.; Peskin, C.S., Implicit second-order immersed boundary methods with boundary mass, Comput. methods appl. mech. engrg., 197, 2049-2067, (2008) · Zbl 1158.74533
[41] Newren, E.P.; Fogelson, A.L.; Guy, R.D.; Kirby, R.M., A comparison of implicit solvers for the immersed boundary equations, Comput. methods appl. mech. engrg., 197, 2290-2304, (2008) · Zbl 1158.76409
[42] Wang, X.S., An iterative matrix-free method in implicit immersed boundary/continuum methods, Comput. struct., 85, 739-748, (2007)
[43] Wolf-Gladrow, D.A., Lattice gas cellular automata and lattice Boltzmann models: an introduction, (2000), Springer Berlin · Zbl 0999.82054
[44] Chen, S.Y.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. rev. fluid mech., 30, 329-364, (1998) · Zbl 1398.76180
[45] Zhu, L.; Tretheway, D.; Petrold, L.; Meinhart, C., Simulation of fluid slip at 3D hydrophobic microchannel walls by the lattice Boltzmann method, J. comput. phys., 202, 181-195, (2005) · Zbl 1061.76072
[46] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases, I: small amplitude process in charged and neutral one-component system, Phys. rev., 94, 511-525, (1954) · Zbl 0055.23609
[47] Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. rev. E, 65, 046308, (2002) · Zbl 1244.76102
[48] Feng, Z.G.; Michaelides, E.E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. comput. phys., 195, 602-628, (2004) · Zbl 1115.76395
[49] Feng, Z.G.; Michaelides, E.E., Proteus: A direct forcing method in the simulations of particulate flows, J. comput. phys., 202, 20-51, (2005) · Zbl 1076.76568
[50] Hindmarsh, A.C.; Brown, P.N.; Grant, K.E.; Lee, S.L.; Serban, R.; Shumaker, D.E.; Woodward, C.S., SUNDIALS: suite of nonlinear and differential/algebraic equation solvers, ACM trans. math. software, 31, 363-396, (2005) · Zbl 1136.65329
[51] Brown, P.N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. sci. stat. comput., 11, 450-481, (1990) · Zbl 0708.65049
[52] Brown, P.N.; Hindmarsh, A.C., Reduced storage matrix methods in stiff ODE systems, J. appl. math. comput., 31, 49-91, (1989) · Zbl 0677.65074
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