## 3 Numerical Modelling
The pressure difference between surfaces \(s1\) and \(s2\) induces movement of the element governed by the following classical second-order equation [22]:  
\[m\ddot{x}+c\dot{x}+kx=F(t) \tag{4}\]  
where  
\(m\) represents the mass of the equivalent BM element;  
\(c\) represents the constant damping of the equivalent BM element;  
\(k\) represents the equivalent stiffness coefficient of the BM element;  
\(F(t)\) represents the force resulting from the pressure difference between surfaces \(s1\) and \(s2\). Equation (1) can be expressed in the following form [22]:  
\[\ddot{x}+2b\dot{x}+\omega_{0}{}^{2}=F_{0} \tag{5}\]  
where  
\[2b=\frac{c}{m} \tag{6}\]  
\[\omega_{0}=\sqrt{\frac{k}{m}} \tag{7}\]  
Figure 6: (**a**) Single spring–damper element of the basilar membrane model: \(k\)—spring element, \(c\)—damper element, \(m\)—mass of the element, \(s1\) and \(s2\)—surface 1 and surface 2. (**b**) Independent spring–damper elements (\(e1\), \(e2\), \(e3\)) constitute the basilar membrane. \[F_{0}=\frac{F(t)}{m} \tag{8}\]  
Then, for the case when \(\omega_{0}>b\), the formulas for the predicted displacement and velocity of the element can be written as follows [22]:  
\[x(t)=\frac{F_{0}}{k}+e^{bt}(C_{1}\text{cos}\omega_{D}t+C_{2}\text{sin}\omega_{D }t) \tag{9}\]  
Figure 4:0  
Figure 4:1  
where the constants C depend on the structural properties  
Figure 4:2  
Figure 4:3  
and where the values are initialized from the previous time step. Analogous expressions can be derived for other cases, where similar conditions hold. As mentioned earlier, the geometry of the model remains unchanged by the displacement of individual elements of the basilar membrane. Only the velocities of these elements are applied as boundary conditions to surfaces \(s1\) and \(s2\) (both velocity-inlet surfaces), which represent the membrane. This means that the fluid flow velocities across surfaces \(s1\) and \(s2\) match the velocities of the elementary spring-damper elements due to displacements. The model was implemented using user-defined functions (UDFs) in ANSYS Fluent. Specifically, three types of macros were utilized:  
* DEFINE_INIT--This macro is executed once during the initialization phase of the simulation. Its purpose is to assign initial values to the variables required for the simulation. * DEFINE_EXECUTE_AT_END--A macro executed at the end of each simulation time step. This macro calculates the pressure difference between surfaces \(s1\) and \(s2\) of the basilar membrane and updates the positions and velocities of the membrane's spring-damper elements. * DEFINE_PROFILE--This macro transfers the velocities of the spring-damper elements to the fluid flow velocities across surfaces \(s1\) and \(s2\). It should be noted that in the discussed model, the oval window and the round window are modelled in the same manner as the basilar membrane. The model presented in this article was validated using experimental results obtained from measurements on the temporal bone with the input data reported in the literature [7]. Unlike other models, this one considers fluid in the cochlea, whereas others typically simplify it to a regular, elastic material. This critical difference makes direct comparisons with other models challenging. However, the simulation results met the experimental findings mentioned above and those reported in study [23].