# Numerical Investigation of Excitation of Various Lamb Waves Modes in Thin Plastic Films
## 2 Finite Elements Models
The 3D model is very complicated and has some disadvantages: a large amount of data is required as input for the mesh used in terms of nodal connectivity and other parameters depending on the problem, and requires longer execution time. Our 3D model data is about 120 GB, with numbers of elements in the model being over 6 million, a calculation time of about 5 h, and where it is impossible to see the S\({}_{0}\) guided wave mode. For this reason, other simulation results were obtained by the 2D modeling method. In the 2D simulation method, we used a finite element size of 15 \(\upmu\)m in the model, which is \(\sim\)4.2 times smaller than in the 3D model and the length of the PVC film is 2.5 times longer than the 3D model. The values of the phase and group velocities of the A\({}_{0}\) and S\({}_{0}\) guided waves modes in PVC film were calculated using the Semi Analytical Finite Element (SAFE) method. The parameters of the clear PVC thin film used for calculations are presented in Table 2[35]. The calculated phase and group velocities versus frequency are shown in Figure 2. From the simulation results it follows that even in the frequency range up to 3.2 MHz, only two modes, symmetrical S\({}_{0}\) and antisymmetrical A\({}_{0}\), may propagate. For numerical simulations the frequency of the excitation signal 150 kHz was selected because at such a frequency only those two modes may propagate. The ultrasound phase velocity of the A\({}_{0}\) mode at the frequency is 232 m/s and the group velocity is 448 m/s are presented by red and blue colors in Figure 2. The ultrasound phase and group velocities of the S\({}_{0}\) mode do not depend on a frequency and are 1595 m/s. In Figure 2 they are indicated by a solid red horizontal line. The ultrasound velocity in air of 343 m/s is presented by a dotted line in Figure 2. Please note that the phase velocity of the A\({}_{0}\) mode is slower than the ultrasound velocity in air up to 240 kHz, therefore its air-coupled excitation by a classical method using deflected ultrasonic transducer looks impossible. However, we shall demonstrate that it is possible using a linear air-coupled array. The waveform and the spectrum of the excitation force are shown in Figures 3 and 4. In the Figures 3 and 4 n.u. represents normalized units. The excitation frequency is \(f=150\) kHz, and the numbers of periods is \(n=3\). \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Advantages and Disadvantages of Simulation Methods**} \\ \hline  & **2D Simulation** & **3D Simulation** \\ \hline The length of the simulated & & quarter of the film \\ PVC film & & 120 mm \(\times\) 120 mm \\ \hline Finite element size & 15 \(\upmu\)m & 65 \(\upmu\)m \\ \hline The simulation step in the & 40 ns & 1 \(\upmu\)s \\ time domain & & \\ \hline Numbers of elements in the & 180,000 & 6,816,600 \\ model & & \\ \hline Calculation time & \(\sim\)20 min & \(\sim\)5 h \\ \hline Data size & \(\sim\)15 GB & \(\sim\)120 GB \\ \hline \multicolumn{3}{l}{**Calculation computer recourses:**} \\ \multicolumn{3}{l}{Processor—Intel (R) Core (TM) i7—2700K CPU @ 3.50 GHz 3.50 GHz;} \\ \multicolumn{3}{l}{Installed RAM—32.0 GB;} \\ \multicolumn{3}{l}{System type—64-bit operating system, \(\times\)64-based processor.} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Advantages and disadvantages of the simulation methods. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Density & \(\rho=1400\) kg/m\({}^{3}\) \\ [PERSON]’s modulus & \(E=2.937\) GPa \\ Poisson’s coefficient & \(\
u=0.42\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of PVC thin film. Figure 3: Excitation signal. Figure 2: Calculated dispersion curves of PVC thin film. The simulation results obtained by the 2D and 3D FEM methods are presented in the following Section 3.