## 2 Bidirectional FSI Numerical Model
### Fluid Domain Solution  
#### 2.2.1 MPS Method  
Numerical modeling of fluid domains using conventional Eulerian frameworks often requires continuous grid reinitialization, which becomes computationally unstable when handling strong nonlinear interfacial interactions and substantial structural deformations. Conversely, Lagrangian-based approaches monitor discrete fluid particle trajectories, inherently ensuring rigorous conservation of mass and momentum--a critical advantage for domains with evolving boundaries. To address the bidirectional FSI between the drill string and drilling fluid, this research adopts the Moving Particle Semi-implicit (MPS) methodology [58; 59], a particle-centric Lagrangian technique that circumvents grid dependency through adaptive particle discretization and robustly resolves extreme deformation scenarios. Assuming that the drilling fluid is incompressible, the spatial fluid is discretized into a finite number of particles with basic physical quantities such as pressure, velocity, and mass. Based on the conservation of mass and momentum, the governing equations in Lagrangian form for the fluid domain can be expressed by Equation (17) [60]. \[\left\{ \begin{array}{l}\frac{D\rho}{DT}=0\\ \frac{D\mathbf{u}}{DT}=-\frac{\
abla P}{\rho}+\mathbf{v}\
abla^{2}\mathbf{u}+ \mathbf{g}\end{array} \right. \tag{17}\]  
where \(\rho\) is the density of the drilling fluid, \(\mathbf{u}\) is the displacement, \(\mathbf{v}\) is the velocity, and \(\mathbf{g}\) is the gravitational acceleration. The interaction between particles is calculated using the weighting function. It is unrealistic to calculate the interaction forces between all particles in the entire drilling fluid domain, and the actual influence between two particles that are far enough apart can be ignored. Therefore, the effective calculation radius of the particles is set as \(r_{\mathrm{e}}\), as shown in Figure 3. The interaction forces are calculated only within \(r_{\mathrm{e}}\). Those beyond the scope will not be considered. The pressure correction of the governing equation is solved implicitly. During discretization, the pressure gradient term is approximated using inter-particle pressure differences among drilling fluid particles, whose formulation is given by Equation (18) [60]. \[\left\langle\
abla P\right\rangle_{i}=\frac{d}{n_{0}}\underset{i\
eq j}{ \sum}\left[\frac{P_{j}-P_{i}(\mathbf{r}_{j}-\mathbf{r}_{i})}{\mathbf{r}_{j}- \mathbf{r}_{i}|\mathbf{r}_{j}-\mathbf{r}_{i}|}W(|\mathbf{r}_{j}-\mathbf{r}_{i }|)\right] \tag{18}\]  
where \(W(|\mathbf{r}_{j}-\mathbf{r}_{i}|)\) is a weighting function, which is used to measure the interaction force between particles. For explicit computation of the viscous and external force terms, the Laplacian operator acting on the velocity difference of particle \(i\) relative to its neighbors is discretized into the form shown in Equation (19) [61]. \[\left\langle\
abla^{2}\mathbf{u}\right\rangle_{i}=\frac{2d}{\lambda n_{0}} \underset{j\
eq i}{\sum}\left(\mathbf{u}_{j}-\mathbf{u}_{i}\right)W(|\mathbf{ r}_{j}-\mathbf{r}_{i}|) \tag{19}\]  
Here, \(d\) represents the spatial dimension, and \(\lambda\) can be expressed as  
\[\lambda=\frac{\underset{j\
eq i}{\sum}\left|\mathbf{r}_{j}-\mathbf{r}_{i} \right|^{2}W(|\mathbf{r}_{j}-\mathbf{r}_{i}|)}{\underset{j\
eq i}{\sum}W(| \mathbf{r}_{j}-\mathbf{r}_{i}|)} \tag{20}\]  
For the basic solution process of time integration, the MPS method couples explicit and implicit solution strategies. Firstly, the velocity field is predicted through explicit calculation, as shown in Equation (21) [58]. Then, the pressure field is solved through  
Figure 3: Effective computational radius of fluid particles. an implicit algorithm, and the update and correction of the velocity field are completed simultaneously. The updated form can be expressed as shown in Equation (22) [58].