The immersed boundary method

TLDR

The immersed boundary (IB) method is a computational framework for simulating fluid–structure interaction, particularly in biological fluid dynamics. This paper investigates the mathematical foundations of the IB method. The IB formulation derives from the principle of least action, coupling Eulerian and Lagrangian variables via a smoothed Dirac delta function on a fixed Cartesian mesh for the fluid and a moving curvilinear mesh for the structure, with data transfer governed by Eulerian/Lagrangian identities and time integration by a second‑order Runge–Kutta scheme. The authors outline current and future research directions and briefly discuss applications of the IB method.

Abstract

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed.