## 4 Verification and Validation
### Validation of the Stability Improvement
#### 4.2.2 Fastening Stiffness  
The fastening force must be added to fix or actuate a certain part of an elastic boundary. Assuming that there is a ghost boundary and all variables defined on it are specified, the actual boundary is connected to this ghost boundary through a set of still springs. In practice, the fastening coefficient should be set to be sufficiently large so that the relative displacement between the actual boundary and the ghost boundary is negligible. However, instability occurs if the fastening coefficient exceeds a threshold. In this section, the effect of the AWI approach on the fastening coefficient is studied by simulating a classic benchmark case: 2D fluid flow around a circular cylinder.  
A circular cylinder with diameter D = 80 is placed in the middle of a computational domain with dimensions \(L\times W=400\times 400\). The cylinder is discretized by 500 Lagrangian points. The Dirichlet boundary condition of velocity \(u_{inlet}=0.1\) is specified at the inlet, and the fluid viscosity is set as \(v=0.1\). All quantities are expressed in lattice units.  
Without loss of generality, the cylinder diameter and inlet velocity are selected as the characteristic length and velocity, respectively. The fastening coefficients are non-dimensionalized according to Equation (15). Under the premise of program stability, Figure 8 shows the attainable maximal fastening coefficient using the AWI approach with different iteration times. The maximal \(\overline{K}_{f}\) is 10 with the explicit method, whereas it linearly increases with the iterations when the proposed method is used. The gradient is approximately 6.5 and the maximal \(\overline{K}_{f}\) can reach 126 when the iteration time is \(N=20\), which is approximately 12 times larger than that of the explicit method.  
Since \(\overline{K}_{f}\) is user-defined, to investigate the accuracy when different values of \(\overline{K}_{f}\) are used, an average error \(E_{d,ij}\) is introduced to quantify the deviation of the actual boundary from the ghost boundary. A relatively small and uniform \(E_{d,ij}\) indicates a smaller introduction error in the fastening force.  
\[E_{d,ij}=\frac{\sum_{N_{b}}\left|\mathbf{X}_{ij}^{a}-\mathbf{X}_{ij}^{g}\right|}{N_{b}} \tag{23}\]  
where \(\mathbf{X}_{ij}\) is the coordinate of the immersed boundary in either the X or Y direction. Superscript a and subscript g indicate the actual and ghost boundaries, respectively.  
Figure 7: Maximal tension stiffness simulated with different iteration steps using the AWI approach.  
The histories of \(E_{d,ij}\) for different \(\overline{K}_{f}\) are shown in Figure 9, where the eight curves correspond to the deviation of the actual boundary from the ghost boundary for \(\overline{K}_{f}\) = 1, 10, 20, and 50 in the X and Y directions. Starting from the zeroth time step, all curves increase; when the time step exceeds 5000, the curves cease to increase except for the one with \(\overline{K}_{f}=1\). Moreover, when \(\overline{K}_{f}\geq 10\), the deviation settles below 0.025, whereas it fluctuates between 0.12 and 0.14 when \(\overline{K}_{f}=1\). Thus, for the problem where a part or all of the boundary is fixed, \(\overline{K}_{f}\) should be larger than 10. It is apparent that if \(\overline{K}_{f}=10\) and the explicit method is used, no other forces can be included because of the instability issue, whereas the AWI approach can maintain stability.