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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: B.H.V. Topping, C.A. Mota Soares
Chapter 12

A Finite Element Approach to the Immersed Boundary Method

D. Boffi*, L. Gastaldi+ and L. Heltai*

*Department of Mathematics "F. Casorati", University of Pavia, Italy
+Department of Mathematics, University of Brescia, Italy

Full Bibliographic Reference for this chapter
D. Boffi, L. Gastaldi, L. Heltai, "A Finite Element Approach to the Immersed Boundary Method", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Progress in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 12, pp 271-298, 2004. doi:10.4203/csets.12.12
Keywords: immersed boundary method, finite element method.

Fluid-structure interaction systems often involve the resolution of the fluid dynamic equations on a moving (that is, time dependent) domain. Several approaches have been considered in order to deal with such problem. Classical ways to overcome the difficulties due to the reconstruction of the mesh at each time step, are the introduction of the arbitrary Lagrangian-Eulerian (ALE) formulation [1,3], and the fictitious domain method, see [2]. Unfortunately, these method are not able to deal with the interaction between fluid and flexible solids with large deformations.

A completely different approach is due to Peskin who developed the immersed boundary method (IBM) (see [6,7]) to study flow patterns around heart valves. The immersed boundary method is designed to handle a flexible boundary immersed in a fluid, hence it is, in particular, suited for biological fluid dynamic problems. As we have already mentioned, the computation requirement to evolve or adapt the mesh becomes considerably expensive in many fluid-structure interaction systems. In the IBM, the structure is thought as a part of the fluid where additional forces are applied, and where additional mass may be localized. Therefore, instead of separating the system in its two components coupled by interface conditions, as it is conventionally done, the incompressible Navier-Stokes equations, with a nonuniform mass density and an applied elastic force density, are used in order to describe the coupled motion of the hydroelastic system in a unified way. The advantage of this method is that the fluid domain can have a simple shape, so that structured grids can be used. On the other hand, the immersed boundary is tipically not aligned with the grid and it is represented using Lagrangian variables, defined on a curvilinear mesh moving through the domain.

The original numerical approach to the IBM is based on the spatial discretization by finite differences. This employes two independent grids, one for the Eulerian variables in the fluid and the other for the Lagrangian variables associated with the immersed boundary. The main difficulty in the spatial discretization consists in the construction of suitable approximation of the Dirac delta function which is used to take into account the interaction equations, see [7]. The temporal discretization that is currently used by Peskin and his coauthors is a second-order accurate Runge-Kutta method, based on the midpoint rule (see, e.g., [4,5]).

Our approach to the discretization of the IBM is completely based upon the finite element method. The aim is to deal with the delta function, which is related to the forces exerted by the immersed structure on the fluid and viceversa, in a variational way. So that there is no need to construct a regularization of the delta function, but its effect is taken into account by its action on the test functions. In particular we present an efficient computational procedure to evaluate the force term along the immersed boundary in the Navier-Stokes equations, which can be used both in the two and three dimentional cases.

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M.-C. Lai and C.S. Peskin, "An immersed boundary method with formal second-order accuracy and reduced numerical viscosity", J. Comput. Phys., 160(2), 705-719, 2000. doi:10.1006/jcph.2000.6483
D.M. McQueen and C.S. Peskin, "Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity", in Mechanics for a New Millennium, Proceedings of the International Conference on Theoretical and Applied Mechanics (ICTAM) 2000. (Aref, H. and J. W. Phillips, eds.), Kluwer Academic Publishers, 2001.
C.S. Peskin, "Numerical analysis of blood flow in the heart", J. Computational Phys., 25(3), 220-252, 1977. doi:10.1016/0021-9991(77)90100-0
C.S. Peskin, "The immersed boundary method", in Acta Numerica, 2002.

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