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On the hyperelastic formulation of the immersed boundary method. (English) Zbl 1158.74523
Summary: The immersed boundary (IB) method is both a mathematical formulation and a numerical method for fluid-structure interaction problems, in which immersed incompressible visco-elastic bodies or boundaries interact with an incompressible fluid. Previous formulations of the IB method were not able to treat appropriately immersed materials of finite, nonzero thickness modeled by general hyper-elastic constitutive laws because of the lack of appropriate transmission conditions between the immersed body and the surrounding fluid in the case of a nonzero jump in normal stress at the solid-fluid interface. (Such a jump does not arise when the solid is comprised of fibers that run parallel to the interface, but typically does arise in other cases, e.g., when the solid contains elastic fibers that terminate at the solid-fluid interface). We present a derivation of the IB method that takes into account in an appropriate way the missing term. The derivation presented in this paper starts from a separation of the stress that appears in the principle of virtual work. The stress is divided into its fluid-like and solid-like components, and each of these two terms is treated in its natural framework, i.e., the Eulerian framework for the fluid-like stress and the Lagrangian framework for the solid-like stress. We describe how the IB method can be used in conjunction with standard formulations of continuum mechanics models for immersed incompressible elastic materials and present some illustrative numerical experiments.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74B20 Nonlinear elasticity
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
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[1] W. Bangerth, R. Hartmann, G. Kanschat, Differential Equations Analysis Library, Technical Reference, http://www.dealii.org. · Zbl 1365.65248
[2] Boffi, D.; Gastaldi, L., A finite element approach for the immersed boundary method, Comput. struct., 81, 8-11, 491-501, (2003)
[3] Boffi, D.; Gastaldi, L., The immersed boundary method: a finite element approach, (), 1263-1266
[4] Boffi, D.; Gastaldi, L.; Heltai, L., A finite element approach to the immersed boundary method, (), 271-298
[5] D. Boffi, L. Gastaldi, L. Heltai, The finite element immersed boundary method: model, stability, and numerical results, in: E. Onate, S. Papadrakakis (Eds.), Computational Methods for Coupled Problems in Science and Engineering, 2005. · Zbl 1186.76661
[6] D. Boffi, L. Gastaldi, L. Heltai, Stability results for the finite element approach to the immersed boundary method, in: Bathe, K. (Ed.), Proceeding of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, 2005, pp. 93-96. · Zbl 1388.76123
[7] D. Boffi, L. Gastaldi, L. Heltai, Stability results and algorithmic strategies for the finite element approach to the immersed boundary method, in: A. Bermudez de Castro, D. Gomez, P. Quintela, P. Salgado (Eds.), Proceedings of ENUMATH 2005 the European Conference on Numerical Mathematics and Advanced Applications, 2006, pp. 557-566. · Zbl 1388.76123
[8] Boffi, D.; Gastaldi, L.; Heltai, L., Numerical stability of the finite element immersed boundary method, Math. models methods appl. sci., 17, 10, 1479-1505, (2007) · Zbl 1186.76661
[9] Boffi, D.; Gastaldi, L.; Heltai, L., On the CFL condition for the finite element immersed boundary method, Comput. struct., 85, 11-14, 775-783, (2007)
[10] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, Springer series in computational mathematics, vol. 15, (1991), Springer-Verlag New York · Zbl 0788.73002
[11] Gurtin, M.E., An introduction to continuum mechanics, Mathematics in science and engineering, vol. 158, (1981), Academic Press Inc. [Harcourt Brace Jovanovich Publishers] New York · Zbl 0559.73001
[12] L. Heltai, The Finite Element Immersed Boundary Method, PhD thesis, Università di Pavia, Dipartimento di Matematica “F. Casorati”, 2006. · Zbl 1388.76123
[13] L. Heltai, On the stability of the finite element immersed boundary method, Computers & Structures, 2007, doi:10.1016/j.compstruc.2007.08.008. · Zbl 1186.76661
[14] Heywood, J.G.; Rannacher, R., Finite element approximation of the nonstationary navier – stokes problem. I. regularity of solutions and second-order error estimates for spatial discretization, SIAM J. numer. anal., 19, 2, 275-311, (1982) · Zbl 0487.76035
[15] Lai, M.-C.; Li, Z., A remark on jump conditions for the three-dimensional navier – stokes equations involving an immersed moving membrane, Appl. math. lett., 14, 2, 149-154, (2001) · Zbl 1013.76021
[16] Liu, C.; Walkington, N.J., An Eulerian description of fluids containing visco-elastic particles, Arch. ration. mech. anal., 159, 3, 229-252, (2001) · Zbl 1009.76093
[17] Liu, W.K.; Kim, D.W.; Tang, S., Mathematical foundations of the immersed finite element method, Comput. mech., 39, 3, 211-222, (2007) · Zbl 1178.74170
[18] McQueen, D.; Peskin, C.; Zhu, L., The immersed boundary method for incompressible fluid – structure interaction, (), 26-30
[19] McQueen, D.M.; Peskin, C.S., Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity, ()
[20] 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
[21] Peskin, C.S., The immersed boundary method, () · Zbl 0428.92010
[22] Peskin, C.S.; McQueen, D.M., A three-dimensional computational method for blood flow in the heart. I. immersed elastic fibers in a viscous incompressible fluid, J. comput. phys., 81, 2, 372-405, (1989) · Zbl 0668.76159
[23] C.S. Peskin, D.M. McQueen, Computational biofluid dynamics, in: Fluid Dynamics in Biology (Seattle, WA, 1991), Contemporary Mathematics, vol. 141, American Mathematical Society, Providence, RI, 1993, pp. 161-186.
[24] Peskin, C.S.; Printz, B.F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 1, 33-46, (1993) · Zbl 0762.92011
[25] 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), (electronic) · Zbl 1051.76016
[26] Stockie, J.M.; Wetton, B.R., Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes, J. comput. phys., 154, 1, 41-64, (1999) · Zbl 0953.76070
[27] Stockie, J.M.; Wetton, B.T.R., Stability analysis for the immersed fiber problem, SIAM J. appl. math., 55, 6, 1577-1591, (1995) · Zbl 0839.35105
[28] 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, 6, 1361-1376, (2004) · Zbl 0760.76067
[29] Wang, X.; Liu, W., Extended immersed boundary method using FEM and RKPM, Comput. methods appl. mech. engrg., 193, 1305-1321, (2004) · Zbl 1060.74676
[30] Zhang, L.; Gerstenberger, A.; Wang, X.; Liu, W., Immersed finite element method, Comput. methods appl. mech. engrg., 193, 2051-2067, (2004) · Zbl 1067.76576
[31] 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, 2, 452-468, (2002) · Zbl 1130.76406
[32] Zhu, L.; Peskin, C.S., Interaction of two flapping filaments in a flowing soap film, Phys. fluids, 15, 7, 1954-1960, (2003) · Zbl 1186.76611
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