# Stability Evaluation of a Simplified Reclaimer Using Fluid-Structure Interaction
## 2 Materials and Methods
### Mathematical Model
#### 2.1.2 Numerical Model
According to [PERSON] [29], the generalized concept of the inertial moment was introduced by [PERSON], using the vibration modes of the bodies to characterize their dynamic behavior. The main formulations used to represent the behavior of mechanical equipment requiring contact such as rolling and sliding are the Newtonian, Lagrange, and Hamiltonian techniques. The models consist of the use of a restitution coefficient, contact stiffness, damping for normal contact, and a friction coefficient for tangential contact [30]. In the past, structural analyses have been linearized and represented in a steady state when subjected to aerodynamic forces, failing to consider the unsteady effect of these forces. Also, complex CFD models have been used to predict physical phenomena influenced by dynamic structural responses, neglecting the deformation caused in the structure. The FSI analysis minimizes the limitations imposed by the former models in one-way or two-way coupling [27]. In the one-way coupling, a converged solution is obtained for one field and then used as a boundary condition for the second field. In the two-way coupling, the fluid and solid equations are solved separately and iterated within each timestep [31]. In the present work, the displacements of the solid are obtained, assuming contact with the fluid and no penetration into the body. The pressure fluctuations are obtained for the fluid motion, which imposes force fluctuations on the solid surface and modifies the motion of the solid. The movement of the solid causes pressure gradients in the fluid and changes its movements. For the interaction between the methods, mesh movement algorithms are required. According to [PERSON] et al. [32], this kind of algorithm has increasingly been used, and the most general technique is the Arbitrary Lagrangian-Eulerian (ALE) technique, which forces the equations to modify the domain at each timestep. In this code [11], there is no distinction between solid and fluid materials; the difference is in the constitutive relations. For the analysis of solids, a Lagrangian technique is used, while for fluids, a Eulerian approach is used. The convection terms are eliminated in a Lagrangian formulation because the material velocity equals the domain velocity, but they are retained in a Eulerian formulation. Constitutive relations defining the Cauchy stress and the heat flux must be stated in order to solve the governing equations. Deformation frequency, the symmetric component of the velocity gradient, is denoted as D (Equation (7)):  
\[\mathrm{D}=\frac{1}{2}\
abla v^{\prime}+\
abla v^{\prime T} \tag{7}\]  
where \(v\) is the velocity deformation. The linearized strain tensor is described in terms of the displacement gradient for small rotations and small strains (Equation (8)):  
\[\mathrm{e}=\frac{1}{2}\
abla\mathrm{u}^{\prime}+\
abla u^{\prime T} \tag{8}\]  
where \(e\) is the strain tensor and \(u\) represents deformation values. There are many well-known constitutive rules for the case of linear geometry, when modest strains and minor rotations are considered. To enable simple interaction with the solids4 FOAM modules, the standard OpenFOAM fluid model implementations have been taken and refactored into the fluid model class design. Laminar and turbulent isothermal incompressible Navier-Stokes formulations have been included within a static domain, and the governing equations take the form of the Navier-Stokes Equations (4)-(6), where the pressure term and the viscous term (first term on the right-hand side of the second equation above) represent the surface force terms in the momentum equation. The difference between the deformed and undeformed configurations, as well as the discrepancy between Eulerian and Lagrangian descriptions, can be overlooked in the case where the strains and rotations are minor; also, the Cauchy and engineering stress tensors coincide. The momentum equation becomes a linear function of the displacement field \(u\) when the definition of engineering stress is a linear function of the displacement vector.