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**Immersed boundary methods.**
*(English)*
Zbl 1117.76049

Lumley, John L. (ed.) et al., Annual review of fluid mechanics. Vol. 37. Palo Alto, CA: Annual Reviews (ISBN 0-8243–0737-2/hbk). Annu. Rev. Fluid Mech. 37, 239-261 (2005).

From the introduction: The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow.
The distinguishing feature of this method was that the entire simulation was carried
out on a Cartesian grid, which did not conform to the geometry of the heart, and a
novel procedure was formulated for imposing the effect of the immersed boundary
(IB) on the flow. Since Peskin introduced this method, numerous modifications and
refinements have been proposed and a number of variants of this approach now exist.
In addition, there is another class of methods, usually referred to as “Cartesian
grid methods”, which were originally developed for simulating inviscid flows with
complex embedded solid boundaries on Cartesian grids (Berger and Aftosmis 1998,
Clarke et al. 1986, Zeeuw and Powell 1991). These methods have been extended to
simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus
have capabilities similar to those of IB methods.

In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli and Zaleski (1999).

Citations:

Peskin CS. 1972. Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis. Physiol., Albert Einstein Coll. Med., Univ. Microfilms. 378:72–30

Berger MJ, Aftosmis MJ. 1998. Aspects (and aspect ratios) of Cartesian mesh methods. Proc. 16th Int. Conf. Numer. Methods Fluid Dyn.

Clarke D, Salas M, Hassan H. 1986. Euler calculations for multi-element airfoils using Cartesian grids. AIAA J. 24:1128–1135

Zeeuw D, Powell K. 1991. An adaptively refined Cartesian mesh solver for the Euler equations. AIAA Pap. 1991–1542

Udaykumar HS, Shyy W, Rao MM. 1996. Elafint: A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries. Int. J. Numer. Methods Fluids 22:691–705

Ye T, Mittal R, Udaykumar HS, Shyy W. 1999. An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156:209–40

Anderson DM, McFadden GB, Wheeler AA. 1998. Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30:139–165

Scardovelli R, Zaleski S. 1999. Direct numerical simulation of free-surface and interfacial flows. Annu. Rev. Fluid. Mech. 31:567–603

For the entire collection see [Zbl 1056.76003].

In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli and Zaleski (1999).

Citations:

Peskin CS. 1972. Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis. Physiol., Albert Einstein Coll. Med., Univ. Microfilms. 378:72–30

Berger MJ, Aftosmis MJ. 1998. Aspects (and aspect ratios) of Cartesian mesh methods. Proc. 16th Int. Conf. Numer. Methods Fluid Dyn.

Clarke D, Salas M, Hassan H. 1986. Euler calculations for multi-element airfoils using Cartesian grids. AIAA J. 24:1128–1135

Zeeuw D, Powell K. 1991. An adaptively refined Cartesian mesh solver for the Euler equations. AIAA Pap. 1991–1542

Udaykumar HS, Shyy W, Rao MM. 1996. Elafint: A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries. Int. J. Numer. Methods Fluids 22:691–705

Ye T, Mittal R, Udaykumar HS, Shyy W. 1999. An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156:209–40

Anderson DM, McFadden GB, Wheeler AA. 1998. Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30:139–165

Scardovelli R, Zaleski S. 1999. Direct numerical simulation of free-surface and interfacial flows. Annu. Rev. Fluid. Mech. 31:567–603

For the entire collection see [Zbl 1056.76003].

### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |