## 5 Mathematical Models
### Models at the Macro-Scale
#### 4.2.2 Immersed Boundary Method  
The Immersed Boundary Method (IBM) can be useful to describe fluid-structure interaction when the structure is intended to be an Immersed Body. In the IBM, the fluid is represented in a Eulerian coordinate system, and the configuration of the Immersed Body is described by a Lagrangian approach [101]. Consider the simulation of incompressible flow around the body in Figure 4a, which is described by Problem 1 where \(\Omega=\Omega_{f}\cup\Omega_{b}\). \(\Omega_{f}\) denotes the surrounding fluid domain occupied by the solid body domain \(\Omega_{b}\) with the boundary denoted by \(\Gamma_{b}\). In an IB method, the boundary condition would be imposed indirectly through a specification of Problem 1. In general, the specification takes the form of a source term (or forcing function) in the governing equations, which reproduces the effect of boundary [102]. The original IBM, created by Peskin [58], is typically appropriate for flows with immersed elastic limits. It was designed for the combined modeling of blood flow and muscle contraction in a beating heart. A collection of massless points moves with the local fluid velocity \(\mathbf{v}\), and these points serve as a Lagrangian tracker of the location of the Immersed Body, represented by a set of elastic fibers. Thus, the coordinate \(X\) of the Lagrangian point is governed by the equation  
\[\frac{\delta X}{\delta t}=\mathbf{v}(X,t) \tag{34}\]  
The effect of the Immersed Body on the surrounding fluid is essentially captured by transmitting the fiber's stress \(\mathbf{F}\) to the fluid through a localized forcing term \(f^{IBM}(x,t)\) in the momentum equations, which is given by  
\[\mathbf{f}^{IBM}(x,t)=\int_{\Omega_{\mathbf{v}}}\mathbf{F}(X,t)\delta(|x-X|)dX, \tag{35}\]  
where \(\delta(\cdot)\) is the Dirac delta function. The Dirac delta function in \(\Re^{n}\) is formally defined by the following:  
\[\delta(x)=\left\{ \begin{array}{cc}+\infty&x=0\\ 0&x\
eq 0\end{array} \right.\]  
and is also constrained to satisfy the identity  
\[\int_{x\in\Re^{n}}\delta(x)dx=1.\]  
The function \(\mathbf{F}\) can be written as  
\[\mathbf{F}(X,t)=-\frac{\wp E}{\wp X}, \tag{36}\]  
where \(E[X]\) is a given functional called _\"the elastic potential energy of the material\"_ in configuration \(X\) and where the notation \(\frac{\wp E}{\wp X}\) is shorthand for the _\"Frechet derivative\"_ of \(E\) with respect to \(X\). Thanks to the use of the Dirac delta function, Equation (34) can be rewritten as  
\[\frac{\delta X}{\delta t}=\int_{\Omega_{\mathbf{v}}}\mathbf{v}(x,t)\delta(|x-X|)dx. \tag{37}\]  
The forcing term is thought to be distributed over a band of cells surrounding each Lagrangian point (see Figure 4b), as the fiber locations typically do not coincide with the nodal points of the Eulerian grid. This distributed force is then imposed on the momentum equations of the surrounding nodes. As a result, a smoother distribution function, represented by \(d\) in this case, effectively replaces the delta function and may be applied to a discrete mesh. The \(d\) function allows for the rewriting of Equations (35) and (37) as follows:  
\[\frac{\delta X}{\delta t} = \int_{\Omega_{\mathbf{v}}}\mathbf{v}(x,t)d(|x-X|)dx, \tag{38}\] \[\mathbf{f}^{IBM}(x,t) = \int_{\Omega_{\mathbf{v}}}\mathbf{F}(X,t)d(|x-X|)dX. \tag{39}\]  
The choice of the distribution function \(d\) is a key ingredient in this method. Several different distribution functions were employed in the past (i.e., see [102] for a list of employed \(d\)). Coupling the IB equations with problems related to the Navier-Stokes incompressible flow requires the reformulation of the Problem 1 as follows:  
**Problem 3**.