Immersed Boundary Method (IBM)

In the ALE method, the mesh conforms to the structural interface (body-conforming), whereas in IBM, the structure is "immersed" on a fixed Cartesian grid. The structural boundary is defined independently of the mesh, and boundary conditions are imposed via a forcing term or interpolation.

It's a method devised by Peskin (1972) for simulating blood flow in heart valves. The original formulation is as follows.

Here, $\mathbf{F}$ is the force density on the Lagrangian interface, $\delta$ is the Dirac delta function, and $\mathbf{X}(t)$ is the interface position.

Peskin's regularized delta function is used. The standard is the 4-point discrete delta function:

where $h$ is the grid spacing and $\phi$ is a smooth kernel function with a support width of 4.

They can be broadly divided into two main categories.

Continuous forcing (Peskin-type) is simple to implement, but the interface blurs by the support width of the delta function. Controlling wall $y^+$ is difficult, making it unsuitable for high-Re turbulent wall-bounded flows.

Discrete forcing (Ghost cell method, Direct forcing method) can represent the interface sharply and achieve near-ALE accuracy for wall boundary layer resolution. However, implementation is complex, especially stable handling of cut-cells for moving interfaces is challenging.

Strictly speaking, the Cut-cell method is a type of IBM, where cells are geometrically cut where the interface crosses them, satisfying conservation laws on the cut cells. It has high conservation and good accuracy, but requires countermeasures for the small cell problem (where cut cell volumes become extremely small). This is addressed with cell merging or flux redistribution.

The Direct Forcing Method (Mohd-Yusof, 1997; Fadlun et al., 2000) is the most widely used. It forces the velocity to the desired boundary condition value at Eulerian cells near the interface.

Here, $\mathbf{u}^*$ is the intermediate velocity without forcing, and $\mathbf{u}_{BC}$ is the velocity that must be satisfied on the interface (for no-slip conditions, the structure's velocity).

It's first-order accurate when the interface is at the cell center. Improved versions achieve second-order accuracy using interpolation based on distance from the interface. In the Ghost cell method, virtual cells (ghost cells) are placed inside the interface, and boundary conditions are imposed via reflection interpolation.

The IBM-FSI coupling flow is as follows.

1. Solve the fluid on the Eulerian grid (including IBM forcing)

2. Interpolate fluid forces on the interface and transfer to the Lagrangian structure

3. Advance the structure in time (update displacement/velocity)

4. Update the IBM mask at the new interface position

5. Return to step 1

The advantages and disadvantages of IBM-FSI compared to ALE are as follows.

Exactly. Problems where structures deform significantly and topology changes, like heart valve opening/closing, parachute deployment, or flag fluttering, are IBM's forte.

It exists, but limitedly.

If strict IBM is needed, using research codes (Nek5000, CaNS, PeleLM, AFiD, etc.) or custom implementation in OpenFOAM is realistic.