## 3 Immersed Boundary Method
### Diffuse Interface IBM
#### 3.1.4 Velocity Correction IBM  
Most of the above-mentioned IBMs have a common characteristic that the boundary force is needed to be obtained during the calculation process. The boundary force \(\mathbf{F}(s,t)\) is then distributed to neighboring grids to obtain the body force of the fluid field \(\mathbf{f}(\mathbf{x},t)\). Although \(F(s,t)\) is modeled by the no-slip boundary condition, such as Equation (21), the final velocity field is computed with \(\mathbf{f}(\mathbf{x},t)\) using Equation (22). The distribution procedure of Equation (7) leads to the failure to satisfy the no-slip boundary condition. The physical phenomenon is streamlined penetration. [PERSON] et al. [29] point out that it is due to pre-determined body force \(\mathbf{f}(\mathbf{x},t)\). To remove this drawback, an iterative procedure [18, 30] or a priori correction [81] is often introduced. [PERSON] and [PERSON] [31, 32] proposed a velocity correction IBM that can accurately satisfy the no-slip boundary condition without the iterative procedure. In this method, the computation of body force \(\mathbf{f}(\mathbf{x},t)\) is replaced by the velocity correction because adding a body force to Equation (2) is equivalent to making a velocity correction in the flow field [29]. Using this method, the body force \(\mathbf{f}(\mathbf{x},t)\) can be seen as a post-determination. A fractional technique is used in this method. Similar to the direct forcing IBM in [22], the body force is neglected initially, and the intermediate fluid velocity \(\mathbf{u}^{*}(\mathbf{x},t)\) is also computed using \(\partial\mathbf{u}/\partial t=\mathbf{rhs}\) first. Then an unknown velocity correction of the fluid field \(\Delta\mathbf{u}(\mathbf{x},t)\) is assumed to obtain the expression of corrected fluid velocity \(\mathbf{u}(\mathbf{x},t)\):  
\[\mathbf{u}=\mathbf{u}^{*}+\Delta\mathbf{u}. \tag{23}\]  
Corresponding to the distribution of boundary force \(\mathbf{F}(s,t)\) to the body force in the fluid \(\mathbf{f}(\mathbf{x},t)\) as Equation (7), a boundary velocity correction \(\Delta\mathbf{U}(s,t)\) is set at the Lagrangian point. It can also be distributed to the Eulerian points using  
\[\Delta\mathbf{u}(\mathbf{x},t)=\int_{\Gamma}\Delta\mathbf{U}(s,t)D(\mathbf{x}-\mathbf{X}(s,t))ds, \tag{24}\]  
\[\Delta\mathbf{u}(\mathbf{x},t)=\sum_{s\in\Gamma}\Delta\mathbf{U}(s,t)D(\mathbf{x}-\mathbf{X}(s,t) )\Delta s, \tag{25}\]  
where Equation (25) is the discrete form, \(\Delta s\) is the arc length of the boundary element. In the intermediate velocity field, the fluid velocity \(\mathbf{u}^{*}(\mathbf{X}(s,t))\) can be computed with Equation (20). The corrected velocity \(\mathbf{u}(\mathbf{X}(s,t))\) at the same position of the Lagrangian point can be computed in the same way. It should equal the boundary velocity \(\mathbf{U}(s,t)\) according to the no-slip boundary condition. Then the relationship reads:  
\[\mathbf{U}(s,t)=\mathbf{u}(\mathbf{X}(s,t))=\sum_{\mathbf{x}\in\Omega}\mathbf{u}(\mathbf{x},t)D(\mathbf{x} -\mathbf{X}(s,t))\mathbf{l}^{3}, \tag{26}\]  
Substituting Equations (20) and (23) into Equation (26), the following equation system is obtained:  
\[\mathbf{U}(s,t)=\mathbf{u}^{*}(\mathbf{X}(s,t))+\sum_{\mathbf{x}\in\Omega}\sum_{s\in\Gamma} \Delta\mathbf{U}(s,t)\Delta sD(\mathbf{x}-\mathbf{X}(s,t))D(\mathbf{x}-\mathbf{X}(s,t))\mathbf{l}^{3}. \tag{27}\]  
This equation system can be rewritten in the form of matrix  
\[A\mathbf{X}=\mathbf{B}, \tag{28}\]  
where \(\mathbf{B}=\mathbf{U}(s,t)-\mathbf{u}^{*}(\mathbf{X}(s,t))\), \(\mathbf{X}\) is the unknown variable to be solved, \(A\) is a matrix. Here, we present the solution process proposed by [PERSON] et al. [33; 34; 35; 82], due to its simplicity. The term \(\Delta\mathbf{U}\Delta s\) is taken as the primary unknowns at the Lagrangian points, which can avoid the calculation of the arc length of a boundary element \(\Delta s\). The matrix \(A\) is of dimension \(N\times N\), where \(N\) is the number of Lagrangian points.