## 4 Numerical Simulation  
A numerical simulation by a finite element method like in previous studies [47; 48; 49; 50; 51; 52; 53] was performed by a simplified model of the experimental setup. The numerical simulation modeled the 1:1:2 mixture as an example among the four mixtures. Therefore, damping ratio, Poisson's ratio and density of the mixture are extracted from previous studies [44; 45; 54; 55] and assigned to the model material content. The two following tasks were defined for the simulation. The first task was determining a frequency-dependent Young's modulus of the model to investigate the capability of the present setup for characterization of the dynamic mechanical properties of the silicone rubber and vocal fold tissue--moreover, investigating possible limitations of the analytical approximation (1) for calculating the frequency-dependent modulus of elasticity, which was originally developed to calculate the static modulus of elasticity. The second task was examining the influence of the pipette-sample boundary condition, pipette tip compression force and the radius of the pipette, whose effects were detected in the measurement. This information helps to improve Equation (1) to be used in the two boundary conditions, while taking into account all the impact factors. The Young's modulus of the model is the criterion to be considered as the true value of the sample modulus of elasticity. Therefore, the first task could be obtained by comparing Young's modulus of the model with the calculated modulus of elasticity from the measurement, at each of the excitation frequencies in the frequency range. To make the Young's modulus and the calculated modulus of elasticity comparable, the predictions from the numerical of the model should be matched with the measured displacement by adjusting the model Young's modulus when the measured pressure applied to the boundary \(\Gamma_{p}\). This process will be explained more in Section 4.3. In the homogeneous computation, a three-dimensional finite element model was defined based on the experimental setup. Since a cubic or cylinder shape of the model in the same dimensions do not affect the results, to reduce the calculation time, an axisymmetric two-dimensional model was created from the \(xz\)-plane cross-section of the 3D model. The \(z\)-axis in the centre of the pipette in the 3D model  
Figure 3: (**a**) schematic picture of the measurement setup; (**b**) drawing of the miniaturized pipettes. is the symmetric axis (r) of the 2D model. With displacement simulated in the 2D and 3D models with the same situation, the results show a difference in range of a few nanometers, which is negligible in comparison with the huge difference in the computation time of the two models. Our explanation in this article is based on the 3D model to have a better understanding of the model behaviour in the simulation while all the simulation results were obtained from the 2D model. The models were prepared and the computations were done using COMSOL Multiphysics (5.4, COMSOL Inc., Stockholm, Sweden) which is a simulation software currently being used for a wide range of applications worldwide. A hollow cylinder with inner radius 1.25 mm and outer radius of 2.25 mm and 1 mm height assumed as the pipette and it was placed on the centre of the upper surface of a cubic in 50 mm \(\times\) 50 mm \(\times\) 10 mm dimension identical to the experimental setup. Figure 4 shows the 2D and 3D models' geometry. In the simulation, two different boundary conditions were defined to investigate the effects of the pipette cross-section friction with the specimen surface. First, the bottom end of the pipette (cylinder) perpendicularly placed on the model top surface, restricted in the \(z\)-direction but free to move in another axis to simulate the situation where the pipette is connected to the sample. To model the 100 mN force which the pipette imposes on the silicone surface in the experimental setup, the 100 mN boundary load was assigned to the cross-section in the \(z\)-direction.