## 2 Methodology
### Mathematical Model
#### 2.2.2 Structural Mechanics Modeling  
Deformation occurs when a structural mechanism experiences fluid flow due to the transfer of energy from fluid flow to solid, resulting in an FSI problem. The water flow (fluid flow) interacts with the rake and the sub-screen models (structural mechanics) at the common interface between the two models. The results of the fluid computation are used to compute the characteristics of solid mechanics. The following assumptions are presented for the structural mechanics model:  
1. The main screen, the sub-screen, and the rake comprise a linear elastic domain that obeys the continuum theory [32].
2. The two-way coupling ALE approach is a more suitable choice than the one-way approach for the present problem, because the considered screening mechanism is expected to undergo oscillatory flow conditions and is therefore susceptible to fatigue cracks, suggesting the need for analysis with greater accuracy [33].  
Below is the governing equation, implemented for the elastic material to determine its solid displacement [10]:  
\[\rho_{\mathrm{s}}\frac{\partial^{2}\mathbf{u_{s}}}{\partial t^{2}}=\frac{ \partial\sigma_{\mathrm{ij}}}{\partial\mathbf{x_{j}}}+\rho_{\mathrm{s}} \mathbf{f_{s}}, \tag{8}\]  
where \(\mathbf{u_{s}}\) denotes the solid displacement (m), \(\sigma_{\mathrm{ij}}\) denotes the stress tensor component within the solid (N/m\({}^{2}\)), \(\rho_{\mathrm{s}}\) denotes the solid density (kg/m\({}^{3}\)), and \(\mathbf{f_{s}}\) represents the body force components exerted on the solid (N/m\({}^{3}\)). The determination of \(\sigma_{\mathrm{ij}}\) emanates from the application of the constitutive equation of a Newtonian fluid. The cumulative force imparted onto the solid boundary by the fluid corresponds to the negative of the reactive force upon the fluid, as expressed by the following equation [34]:  
\[\mathbf{f_{s}}=\mathrm{n}\cdot\{-\mathrm{p}\mathbf{I}+\left(\mathrm{u}\Big{(} \
abla\mathbf{u_{f}}+(\
abla\mathbf{u_{f}})^{\mathrm{T}}\Big{)}\right)-\frac{ 2}{3}\mathrm{u}(\
abla\cdot\mathbf{u_{f}})\mathbf{I}\}, \tag{9}\]  
where \(\mathbf{u_{f}}\) represents the velocity vector of the fluid medium in meters per second (m/s), while p denotes the fluid pressure (Pa). The identity matrix is represented by the symbol \(\mathbf{I}\). Additionally, the parameter \(\mathbf{f_{s}}\) pertains to the body force exerted upon the solid substance (N/m\({}^{3}\)). n designates the outward normal vector, which characterizes the orientation that is perpendicular to the boundary under consideration. The energy stored in the filtering mechanism due to the structural deformation is measured in the form of the strain energy density function W as follows [34]:  
\[\mathrm{s_{ij}}=\frac{\partial W}{\partial\epsilon_{ij}}\text{ where }\mathrm{i}, \mathrm{j}\,\epsilon\{1,2,3\}, \tag{10}\]  
where \(\mathrm{s_{ij}}\) and \(\epsilon_{ij}\) are the second Piola-Kirchhoff stress and the Green-Lagrange strain tensor components, respectively [35; 36]. Lastly, a transformation of the force between the two coordinate systems is necessary, because the solid mechanics interfaces are defined in the material (undeformed) frame, whilst the Navier-Stokes equations are solved in the spatial (deformed) frame. Hence, a coordinate transformation is performed in the ALE model as follows:  
\[\mathbf{f_{f}}=\mathbf{f_{s}}\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}\mathbf{V}}, \tag{11}\]  
where \(\mathbf{f_{f}}\) is the force on the fluid (N/m\({}^{3}\)), \(\mathbf{f_{s}}\) is the body force acting on the solid (N/m\({}^{3}\)), and dv and dV are the mesh element scale factors for the spatial and material (reference) frames, respectively. In the modeling of the structural mechanics, linear elastic materials are assumed for the WSM components. Boundary loads of fluid stresses are applied at the WSM components to simulate the interaction with fluid forces.