×

zbMATH — the first resource for mathematics

The blob projection method for immersed boundary problems. (English) Zbl 0962.74078
Summary: We present a new finite difference method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two-dimensional domain. The method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled. These are described by a collection of regularized point forces, and the velocity field they induce is computed directly on a regular Cartesian grid via a smoothed dipole potential. We present comparisons between this method and the immersed boundary method of C. S. Peskin and D. M. McQueen [ibid. 81, No. 2, 372-405 (1989; Zbl 0668.76159)]. The results show that the method proposed here preserves volumes better and has a higher order of convergence.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74K15 Membranes
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, C.R., A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. comput. phys., 62, 111, (1986) · Zbl 0575.76031
[2] J. T. Beale, and, M.-C. Lai, A method for computing nearly singular integrals, submitted for publication. · Zbl 0988.65025
[3] Beale, J.T.; Majda, A., High order accurate vortex methods with explicit velocity kernels, J. comput. phys., 58, 188, (1985) · Zbl 0588.76037
[4] T. F. Buttke, Velocity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow, in Vortex Flows and Related Numerical Methods, edited by J. T. Beale, G.-H. Cottet, and S. Huberson, NATO ASI Series C, Vol. 395Kluwer Academic, Dordrecht/Norwel, MA, 1993, pp. 39-57. · Zbl 0860.76064
[5] Chorin, A.J., On the convergence of discrete approximations to the navier – stokes equations, Math. comput., 23, 341, (1969) · Zbl 0184.20103
[6] Cortez, R., An impulse-based approximation of fluid motion due to boundary forces, J. comput. phys., 123, 341, (1996) · Zbl 0847.76064
[7] Cortez, R.; Varela, D.A., The dynamics of an elastic membrane using the impulse method, J. comput. phys., 138, 224, (1997) · Zbl 0910.73035
[8] Shu, C.-W., A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow, (1992)
[9] Fauci, L.J., Interaction of oscillating filaments—A computational study, J. comput. phys., 86, 294, (1990) · Zbl 0682.76105
[10] Fauci, L.J.; Fogelson, A.L., Truncated Newton methods and the modeling of complex immersed elastic structures, Commun. pure appl. math., 46, 787, (1993) · Zbl 0741.76103
[11] Fauci, L.J.; McDonald, A., Sperm motility in the presence of boundaries, Bull. math. biol., 57, 679, (1995) · Zbl 0826.92017
[12] Fauci, L.J.; Peskin, C.S., A computational model of aquatic animal locomotion, J. comput. phys., 77, 85, (1988) · Zbl 0641.76140
[13] Fogelson, A.L., A mathematical model and numerical method for studying platelet adhesion and aggregation during blood cloting, J. comput. phys., 56, 111, (1984) · Zbl 0558.92009
[14] O. H. Hald, Convergence of vortex methods, in Vortex Methods and Vortex Motion, edited by K. E. Gustafson and J. A. SethianSIAM, Philadelphia, 1991, pp. 33-58.
[15] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., Removing the stiffness from interfacial flows with surface tension, J. comput. phys., 114, 312, (1994) · Zbl 0810.76095
[16] LeVeque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019, (1994) · Zbl 0811.65083
[17] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220, (1977) · Zbl 0403.76100
[18] 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, 372, (1989) · Zbl 0668.76159
[19] C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, in Biological Fluid Dynamics, edited by C. P. Ellington and T. J. Pedley, Symposia of the Society for Experimental Biology, Vol. 19Cambridge Univ. Press, Cambridge, UK, 1995, pp. 265-276.
[20] Peskin, C.S.; Printz, B.F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 33, (1993) · Zbl 0762.92011
[21] Rosar, M.E., A three-dimensional computer model for fluid flow through a collapsible tube, (1994)
[22] Stockie, J., Analysis and computation of immersed boundaries, with application to pulp fibres, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.