## 2 Methodology
### Mathematical Model  
The interaction of the water flow with the rake and sub-screen models occurs at the interface shared by the two models. This phenomenon is called FSI, and the flow of the sewage water through the screening mechanism is an FSI problem. The present study uses the ALE approach with two-way coupling to solve such a problem. In the ALE approach, first, the fluid flow is solved using the Navier-Stokes equation. The result obtained from the fluid flow solution is then used to calculate the structural deformation, and the fluid flow is solved using the Eulerian method. The coordinate system in the Eulerian method is fixed considering the computational domain. This frame of reference is called the spatial frame. The obtained solution is then fed into the solid mechanical study using the Lagrangian approach. The equations in the Lagrangian method are stated for small individual volumetric elements of the rigid body during deformation. Thus, the associated frame of reference moves with the individual particles during the rigid body deformation. The coordinate system moves with the rigid body as it deforms; thus, the location of the rigid object remains stationary, considering the Lagrangian coordinate system. This frame is called the material frame. The object appears undeformed in the Lagrangian frame [22; 23]. The aforementioned theoretical approach may be used to address the FSI-based problems via two distinct forms of coupling: one-way and two-way coupling. The one-way coupling technique only accounts for the impacts of water flow on the rake and sub-screen models. In this technique, the structural impacts of the rake and sub-screen models do not have any influence on the flow of water. The structural features of the rake and sub-screen models are influenced by the water flow in the two-way coupling technique. Therefore, the water flow field is likewise influenced by the structural features of these solid models. The present methodology employs an iterative loop in order to facilitate the transfer of attributes from one domain to another. The fluid stresses derived from the sewage flow are \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **[PERSON] et al. [7]** & **Present Study** \\ \hline Channel width & 4.2 m & 3.40 m upstream, 3.11 m downstream \\ Thickness of the sub-screen bar & 0.06 m & 0.075 m \\ Sub-screen position & [0\({}^{\circ}\) 15\({}^{\circ}\)] & fully lowered, partially lowered \\ Sub-screen height & 1.11 m & 0.95 m \\ Number of bars on the main screen & N/A & 37 \\ Number of bars on the sub-screen & N/A & 39 \\ Number of rake pins & N/A & 36 \\ Rake position & \{15\({}^{\circ}\), 105\({}^{\circ}\)] & \{-75\({}^{\circ}\), 0\({}^{\circ}\)] \\ Rake length & 0.33 m & 0.34 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of parameters of the present study with [PERSON] et al. [7]. used to generate mappings for the rake and sub-screen models. Subsequently, the ensuing displacements occurring in the structures are then sent back to the water flow. Either one of the one- or two-way coupling approaches may be used to conduct the present study. To identify the more suitable approach between the two, we consider the operating conditions of the WSM and the availability of computational tools. In considering the operating conditions, we realize that the turbulence in the water flowing through the sewage, coupled with the fluctuation in the flow velocity and the disturbance due to the rake movement against the water flow, would induce oscillatory loading in the screening mechanism. Owing to this oscillatory loading, the structure is susceptible to the initiation and propagation of fatigue cracks. Once a fatigue crack is initiated, it may grow gradually with each loading cycle and cause structural damage over several cycles. Failing to consider the possibility of fatigue can be catastrophic, especially in structures that are susceptible to oscillatory loading conditions.