## 3 Numerical Modelling  
A simplified 3D model for numerical calculations of cochlea dynamics, including crucial elements such as the oval window, the round window, and the basilar membrane between the scala vestibuli and scala tympani, has been developed (Figure 4). The key feature of the model is its representation of the basilar membrane as two independent surfaces (\(s1\) and \(s2\)) separated by a gap, with each of these surfaces discretized into independent and not connected finite sections. The elements are arranged in pairs, which means each element on one surface has a corresponding one on the other surface. In this simplified model, the external auditory canal and the ossicles are not taken into consideration. Hence, there is no comparison to the open and closed canals from the previous section. Figure 3: FRF ratios (\(H_{TV(RWM)}/H_{TV(SF)}\)) measured for the RWM (\(H_{TV(RWM)}\)) and the stages footplate (\(H_{TV(SF)}\)) from experiments with open (red) and closed (blue) canals. Black points represent outcomes from CFD simulations. The dynamic behaviour of the cochlea system is simulated using a single CFD solver, eliminating the need for fluid-structure fully coupled calculations [19; 20; 21]. In the proposed conceptual model, membrane mesh deformation is neglected, based on the assumption that the transverse displacements induced by pressure differences are minimal. Instead, the basilar membrane is modelled as a collection of independent elastic-damping elements, each behaving like a classical mass-spring-damper system (see Figure 5). Any individual element has assigned distinct physical properties, and its motion is governed by pressure differences across the membrane surfaces. In this approach, the model does not account for geometric changes caused by potential deformations or movements. Instead, the velocities of the basilar membrane elements are directly applied as boundary conditions to the fluid domain, effectively transforming the actual displacements of the elastic-damping elements into fluid velocities. This new model, implemented using user-defined functions (UDFs) in ANSYS Fluent, streamlines the simulation process by significantly reducing computational effort while still capturing the essential dynamics of fluid flow and pressure distribution within the cochlea. The equivalent parameters of the proposed basilar membrane model were determined through additional, independent finite element method simulations. In these calculations, the membrane was subjected to a constant uniform pressure, causing it to deform. The resulting deformation was referenced to a fixed coordinate system. Based on the known de  
Figure 4: The inner ear model with visible surfaces (\(s1\) and \(s2\)) representing the equivalent of the basilar membrane. Figure 5: Trend of basilar membrane stiffness (\(k\)) relative to position from the basilar membrane’s origin. formation of each element and its force load (calculated by multiplying pressure and surface area), the stiffness coefficient \(k\) of each individual element was determined (Figure 5). This stiffness was then used in the cochlea model simulation, where the membrane was replaced by two velocity-inlet surfaces, and its deflection was represented by corresponding velocity values at these inlets (Figure 6). With this approach, each membrane element moves independently from its neighbours and has uniquely assigned structural properties. A conceptual model of any generic basilar membrane element is represented in Figure 6, and involves a membrane section, divided into two independent surfaces, \(s1\) and \(s2\), as well as a spring and damper connected in parallel. These elements, referred to as spring-damper components, are modelled as springs in parallel with dampers (Figure 6). Each individual element is assigned a mass, stiffness coefficient, damping constant, and a corresponding section of the basilar membrane, divided into two independent surfaces, \(s1\) and \(s2\).