## 3 Immersed Boundary Method
### Sharp Interface IBM  
The boundaries in all the above diffuse interface IBMs are smeared across several grid cells, which depends on the choice of the Dirac delta function in Figure 3. A series of sharp interface IBMs is developed to eliminate the diffuse feature. The hybrid Cartesian-IBM proposed by [PERSON] et al. [43] is a representative sharp interface IBM and has been widely applied in the simulations of bio-inspired motions. This method removes the momentum forcing term from the computational process, which means that the governing equations of fluid flow are the same as Equations (1) and (2) but with the term \(f=0\) in Equation (2). In this method, the fluid field is solved by a well-developed Navier-Stokes solver. Different from the above-mentioned diffuse interface methods, Lagrangian points are not required to represent the boundary surface, which is inspired by [PERSON] [74] and [PERSON] et al. [75]. The ghost-cell methodology is the key step to finding three types of points, as shown in Figure 6, where the ghost cells (GC) are defined as the ones that locate inside the solid body and have at least one neighboring node in the fluid domain, every ghost-cell has a symmetric point, called the image point (IP), and the intersection on the immersed boundary is the boundary intercept (BI). To satisfy the boundary condition on the interface in the vicinity of each ghost cell, it is required to determine the flow properties at the ghost cells using appropriate interpolation schemes. Figure 5: The average CPU time consumption by different IBMs in each time step versus the number of Lagrangian points. Reproduced from [83]. Note that there are four neighboring cell points in 2D cases, such as the four corners of the shadow square in Figure 6, and eight points in 3D cases. Here, we present the method in 3D form. For a flow property \(\varphi\) (denoting velocity \(\mathbf{u}\) and pressure \(p\)), a trilinear interpolant is given as:  
\[\varphi(x,y,z)=C_{1}xyz+C_{2}xy+C_{3}yz+C_{4}zx+C_{5}x+C_{6}y+C_{7}z+C_{8}, \tag{32}\]  
where the coefficients \(C_{i}\) are determined by the value of \(\varphi\) at the eight points. If the ghost-cell is included in these points surrounding the image point, it should be replaced by the boundary intercept. Then the values at the image points can be computed using  
\[\varphi_{IP}=\sum_{i=1}^{8}\beta_{i}\varphi_{i}, \tag{33}\]  
where \(\beta_{i}\) is computed by the coefficients \(C_{i}\) as well as \((x,y,z)_{IP}\) according to Equation (32). Then the Dirichlet boundary condition is applied for the velocity  
\[\varphi_{GC}+\varphi_{IP}=2\varphi_{BI}. \tag{34}\]  
The Neumann boundary condition is applied for the pressure  
\[\varphi_{GC}-\varphi_{IP}=-\Delta I\left(\frac{\delta\varphi}{\delta n} \right)_{BI}, \tag{35}\]  
where \(\delta\varphi/\delta n\) is the second-order central difference, \(\Delta I\) is the gradient distance from GC to IP. [PERSON] et al. [43] validated the second-order global and local accuracy by simulating the case of flow past a circular cylinder. The results are shown in Figure 7, where the error norms show nearly second-order convergence and indicate the hybrid Cartesian-IBM is globally and locally second-order accurate. Figure 6: The 2D schematic of ghost-cell methodology: GC for ghost-cell, IP for image point, BI for boundary intercept. Reproduced from [43]. This method is improved by [PERSON] et al. [44] and [PERSON] et al. [45], aiming to suppress the numerical oscillations caused by the motion of the immersed boundary. These methods reserve the sharp characteristics of interfaces and improve local accuracy. For bio-inspired motions, they are developed to simulate the flight of the dragonfly [43], hummingbird [44], and cicada [45]. Later, these series of sharp interface IBMs are extended to study more bio-fluid problems [90; 91; 92; 93; 94; 95; 96; 97; 98]. Here, we summarize the numerical characteristics of the above-mentioned IBMs in Table 1. With the development of the IBM, the applicability to rigid structures has been completely achieved.