The Immersed Boundary Method (IBM) has been introduced by Peskin in the 70's in order to model and approximate fluid-structure interaction problems related to the blood flow in the heart. The original scheme makes use of finite differences for the discretization of the Navier-Stokes equations. Recently, a finite element formulation has been introduced which has the advantage of handling the presence of the solid (modeled via a Dirac delta function) in a more natural way. In this paper we review the finite element formulation of the IBM focusing, in particular, on the choice of the finite element spaces in order to guarantee a suitable mass conservation. Moreover, we present some links with the fictitious domain method.
@article{BUMI_2012_9_5_3_711_0,
author = {Daniele Boffi},
title = {The Immersed Boundary Method for Fluid-Structure Interactions: Mathematical Formulation and Numerical},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {5},
year = {2012},
pages = {711-724},
zbl = {1290.76060},
mrnumber = {3051741},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2012_9_5_3_711_0}
}
Boffi, Daniele. The Immersed Boundary Method for Fluid-Structure Interactions: Mathematical Formulation and Numerical. Bollettino dell'Unione Matematica Italiana, Tome 5 (2012) pp. 711-724. http://gdmltest.u-ga.fr/item/BUMI_2012_9_5_3_711_0/
[1] - - , Fluid-structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (14) (2008), 7027-7051. | MR 2435441 | Zbl 1140.74010
[2] - - , Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect, Comput. Methods Appl. Mech. Engrg., 197 (49-50) (2008), 4216-4232. | MR 2463662 | Zbl 1194.74058
[3] - - , Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM J. Sci. Comput., 30 (4) (2008), 1778-1805. | MR 2407141 | Zbl 1368.74021
[4] - , Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math., 33 (2) (1979), 211-224. | MR 549450 | Zbl 0423.65058
[5] , Stability of higher order triangular Hood-Taylor methods for the stationary Stokes equations, Math. Models Methods Appl. Sci., 4 (2) (1994), 223-235. | MR 1269482 | Zbl 0804.76051
[6] , Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal., 34 (2) (1997), 664-670. | MR 1442933 | Zbl 0874.76032
[7] - - - , Local mass conservation of Stokes finite elements, To appear in J. Sci. Comput. (2011). | MR 2948699 | Zbl 1264.74259
[8] - - , Finite element approach to immersed boundary method with different fluid and solid densities, To appear in Math. Models Methods Appl. Sci. (2011). | MR 2864640 | Zbl 1242.76190
[9] - - - , In preparation.
[10] - , In preparation.
[11] - , A finite element approach for the immersed boundary method, Comput. & Structures, 81 (8-11) (2003), 491-501. In honor of Klaus-Jürgen Bathe. | MR 2001876
[12] - - , Numerical stability of the finite element immersed boundary method, Math. Models Methods Appl. Sci., 17 (10) (2007), 1479-1505. | MR 2359913 | Zbl 1186.76661
[13] - - , On the CFL condition for the finite element immersed boundary method, Comput. 85 (11-14) (2007), 775-783. | MR 2308727
[14] - - - , On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Engrg., 197 (25-28) (2008), 2210-2231. | MR 2412821 | Zbl 1158.74523
[15] - , Multigrid techniques: 1984 guide with applications to fluid dynamics, volume 67 of Classics in applied mathematics, SIAM, Philadelphia, 2011. | MR 3396211 | Zbl 1227.65121
[16] - , Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991. | MR 1115205 | Zbl 0788.73002
[17] - - , Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Engrg., 194, (42-44) (2005), 4506-4527. | MR 2157973 | Zbl 1101.74027
[18] - , The inf-sup test, Comput. & Structures, 47 (4-5) (1993), 537-545. | MR 1224095 | Zbl 0780.73074
[19] - - , Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions, M2AN Math. Model. Numer. Anal., 37 (4) (2003), 601-616. | MR 2018432 | Zbl 1118.74315
[20] - - , editors, Cardiovascular mathematics, volume 1, of MS&A. Modeling, Simulation and Applications, Springer-Verlag Italia, Milan, 2009. Modeling and simulation of the circulatory system. | MR 2488002 | Zbl 1300.92005
[21] - - , A fictitious-domain method with distributed multiplier for the Stokes problem, In Applied nonlinear analysis, pages 159-174. Kluwer/Plenum, New York, 1999. | MR 1727447 | Zbl 0954.35127
[22] - - , A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Engrg., 111 (3-4) (1994), 283-303. | MR 1259864
[23] - - , Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies, Comput. Methods Appl. Mech. Engrg., 151 (1-2) (1998), 181-194. Symposium on Advances in Computational Mechanics, Vol. 3 (Austin, TX, 1997). | MR 1625430 | Zbl 0916.76052
[24] - - , Fictitious domain method for the Dirichlet problem and its generalization to some flow problems, In Finite elements in fluids, Part I, II (Barcelona, 1993), pages 347-368, Centro Internac. Métodos Numér. Ing., (Barcelona, 1993). | MR 1292053 | Zbl 0876.76060
[25] - - - , A new finite element for Boussinesq fluids. In Pro. Third Int. Conf. on Finite Elements in Flow Problems, pages 204-215. Wiley, New York, 1980. | Zbl 0447.76026
[26] , The effect of pressure approximation on finite element calculations of compressible flows. In Numerical Methods for Fluid Dynamics, pages 359-374. Academic Press, and edition, 1982.
[27] - - - , Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (18) (2009), 6916-6937. | MR 2567876 | Zbl 1261.76056
[28] , On the Stability of the Finite Element Immersed Boundary Method, Comput. & Structures, 86 (7-8) (2008), 598-617.
[29] - , Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Engrg., 190 (24-25) (2001), 3039-3067. | Zbl 1001.74040
[30] - - , Algorithms for strong coupling procedures, Comput. Methods Appl. Mech. Engrg., 195, (17-18) (2006), 2028-2049. | MR 2202913 | Zbl 1142.74050
[31] - , Partitioned strong coupling algorithms for fluidstructure interaction, Comput. Struct., 81 (2003), 1277-1286.
[32] HENRI - , Interactions fluides-structures, volume 23 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1992. | MR 1180076 | Zbl 0754.73071
[33] , Numerical analysis of blood flow in the heart, J. Computational Phys., 25, (3) (1977), 220-252. | MR 490027 | Zbl 0403.76100
[34] , The immersed boundary method, Acta Numer., 11 (2002), 479-517. | MR 2009378 | Zbl 1123.74309
[35] - , 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) (1989), 372-405. | MR 994353 | Zbl 0668.76159
[36] , Local mass conservation and -discretizations of the Stokes problem, Houston J. Math., 20 (1) (1994), 115-127. | MR 1272565 | Zbl 0803.65105
[37] - , Stability of the finite elements and for stationary Stokes equations, Comput. 84 (1-2) (2005), 70-77. | MR 2199741
[38] - , Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér., 19 (1) (1985), 111-143. | MR 813691 | Zbl 0608.65013
[39] - , A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. 1 (1) (1973), 73-100. | MR 339677 | Zbl 0328.76020
[40] , Locally mass-conserving Taylor-Hood elements for two- and three-dimensional flow, Internat. J. Numer. Methods Fluids, 11 (3) (1990), 341-353. | MR 1065219 | Zbl 0709.76083
[41] - - , The free surface flow of Newtonian and non-Newtonian fluids trapped by surface tension, Internat. J. Numer. Methods Fluids, 8 (9) (1988), 1011-1027. | MR 961777