## 4 Numerical Simulation
### Boundary Condition  
Four types of boundary conditions, \(\Gamma_{p}\), \(\Gamma_{D_{z}}\), \(\Gamma_{D}\), \(\Gamma_{N}\), were defined in the model for different surfaces as shown in Figure 4. The \(\Gamma_{p}\) is the boundary of the model surface inside the pipette, the \(\Gamma_{D_{z}}\) corresponds to the pipette surface cross-section boundary condition, \(\Gamma_{D}\) is the boundary of the bottom side of the model and all the other boundaries are shown with \(\Gamma_{N}\). The amplitude of the measured pressure applied to the boundary \(\Gamma_{p}\) in which the pipette wall surrounds the surface. The relation (6) denotes the Cauchy stress tensor on the boundary \(\Gamma_{p}\). The Cauchy stress tensor is a function of \(p_{m}\) representing the excitation pressure on the surrounded area:  
\[[\sigma]^{T}\mathbf{n}=p_{m}\mathbf{n}\quad\text{on}\quad\Gamma_{p}\times(0, T). \tag{6}\]  
Numerical simulation is conducted under the two following approaches regarding the pipette-sample connection condition. First, the boundary \(\Gamma_{D_{z}}\), which corresponds to the cross-section of the pipette tip wall and the surface of the sample, is restricted in the \(z\)-direction but free to move in the \(x\)-\(y\) plane. This boundary condition is called connected pipette (the pipette touches the surface) in this paper:  
\[u_{z}=0\qquad\text{on}\quad\Gamma_{D_{z}}\times(0,T).\]  
The 100 mN constant load in the \(z\)-direction is assigned to the boundary \(\Gamma_{D_{z}}\) to reproduce the case in the experimental setup in which the pipette applies force on the silicone surface. \(F_{c}\) represents the 100 mN constant contact force:  
\[[\sigma]^{T}\mathbf{n}=F_{c}\mathbf{n}\quad\text{on}\quad\Gamma_{D_{z}} \times(0,T).\]  
Second, the \(\Gamma_{D_{z}}\) is free to move in the \(x\)-\(y\)-\(z\) plane to model the gap in the experimental setup. We will call this boundary condition a pipette with the gap condition in this paper:  
\[u_{z}\quad\text{is}\quad\text{free}\quad\text{on}\quad\Gamma_{D_{z}}\times(0, T).\]Model movement is completely restricted leaving no degree of freedom from the bottom side:  
\[\mathbf{u}=0\quad\text{on}\quad\Gamma_{D}\times(0,T).\]  
All the other boundaries (\(\Gamma_{N}\)) are free in all directions; therefore, homogeneous edge conditions are applied:  
\[[\sigma]^{T}\mathbf{n}=0. \tag{7}\]  
Based on Equation (5), the dynamics of a single degree freedom systems with viscous damping are described by the following equation of motion in matrix form:  
\[\mathbf{M}\frac{\partial^{2}}{\partial t^{2}}\mathbf{u}+\mathbf{K}\mathbf{u}= \mathbf{f}. \tag{8}\]  
In Equation (8), \(\mathbf{u}\) is the discrete displacement, \(\mathbf{M}\) is the mass matrix and \(\mathbf{K}\) denotes the stiffness matrix. The excitation force per unit area enters the equation by the term \(\mathbf{f}\) on the right-hand side. Damping: the damping of the linear elastic model is introduced via the Rayleigh damping in the simulation (see e.g., [56; 59; 60]). Since we will just consider steady-state vibrations and not any transient behaviour, Rayleigh damping is a simple way of generating a damping matrix as a pure linear combination of the mass and stiffness matrices. The damping can be added to Equation (8) by introducing damping matrix \(\mathbf{C}\). The damping matrix consists of mass matrix \(\mathbf{M}\) and the stiffness matrix \(\mathbf{K}\). In Equation (9), \(\alpha_{M}\) denotes the mass proportional and \(\alpha_{K}\) the stiffness proportional damping coefficients, giving:  
\[\mathbf{C}=\alpha_{M}\mathbf{M}+\alpha_{K}\mathbf{K}. \tag{9}\]  
Equation (10) made by combining the matrix (9) into Equation (8) which numerical simulation computations are based on:  
\[\mathbf{M}\frac{\partial^{2}}{\partial t^{2}}\mathbf{u}+\mathbf{C}\frac{ \partial}{\partial t}\mathbf{u}+\mathbf{K}\mathbf{u}=\mathbf{f}. \tag{10}\]  
Harmonic analysis performed in the frequency range to meet the measurement where we measured the steady state of the exciting pressure and mechanical displacement of the sample.