# Numerical Investigation of Excitation of Various Lamb Waves Modes in Thin Plastic Films
## 2 Finite Elements Models  
Propagation of the guided A\({}_{0}\) and S\({}_{0}\) mode in the PVC film sample was simulated by the ABAQUS software package (Dassault Systemes, Johnston, Rhode Island, United States of America). The ABAQUS program was chosen for its high level of detail, manual design of the work piece, possibility of configuring the materials, and the very fine control of mesh; however, the ABAQUS program has no support for any materials and takes time to \"setup\" simulations as the user must manually set many simulation parameters. For the modelling of propagation of A\({}_{0}\) and S\({}_{0}\) modes in thin plastic films, two 2D and 3D calculation methods were chosen [30; 31; 32; 33]. Modeling of elastic wave propagation was carried out by solving the following equation:  
\[[\mathbf{M}]\Big{\{}\ddot{\mathbf{U}}\Big{\}}+[\mathbf{C}]\Big{\{}\dot{\mathbf{U}}\Big{\}}+[ \mathbf{K}]\{\mathbf{U}\}\ =\ \{\mathbf{F}(t)\}, \tag{1}\]  
where \([\mathbf{M}]\) is the structural mass matrix, \([\mathbf{C}]\) is the element damping matrix, \([\mathbf{K}]\) is the structural stiffness matrix, \(\{\mathbf{U}\}\) is the displacement vector, and \(\{\mathbf{F}\}\) is the load vector. The Abaqus Explicit software uses a central difference method to integrate the equation of motion in time [25; 34]:  
\[\{\mathbf{U}_{t+\Delta t}\}\ =\ \big{[}\widehat{\mathbf{M}}\big{]}^{-1}\bigg{[}\{\mathbf{F}(t )\}-\bigg{(}[\mathbf{K}]-\frac{2}{\Delta t^{2}}[\mathbf{M}]\bigg{)}\{\mathbf{U}_{t}\}- \big{[}\widetilde{\mathbf{M}}\big{]}\{\mathbf{U}_{t-\Delta t}\}\bigg{]}, \tag{2}\]  
where \(\Delta t\) is the time step and  
\[\big{[}\widehat{\mathbf{M}}\big{]}\ =\ \frac{1}{\Delta t^{2}}[\mathbf{M}]+\frac{1}{2 \Delta t}[\mathbf{C}], \tag{3}\]  
\[\big{[}\widetilde{\mathbf{M}}\big{]}\ =\ \frac{1}{\Delta t^{2}}[\mathbf{M}]-\frac{1}{2 \Delta t}[\mathbf{C}]. \tag{4}\]  
Three-dimensional models are in practice limited to small size models and low frequencies due to the required computational power to numerically solve the wave's propagation tasks. They also require relatively time-consuming post processing before the results can be obtained. However, there is a significant difference in the computational time between two-dimensional (2D) FE models and more accurate 3D models. First of all, a 3D method was used to simulate the propagation of guided waves excited by one element with the excitation zone of 5 \(\times\) 1 mm, the center of which is at \(x,y=0\). The calculations were performed for polyvinyl chloride (PVC, London, UK). Dimensions of the PVC sample selected for simulation were 240 \(\times\) 240 mm. For reduction of the computational time, only a quarter of the film with symmetry boundary conditions was modeled by the 3D method. The modeled structure of one excitation zone is shown in Figure 1a. The simulation step in the time domain was set \(dt=1\) us. The strip-like piezoelectric element radiates an ultrasonic wave, which propagates via the small air gap and excites a guided wave in the thin plastic film. The excitation was introduced as the uniformly distributed 1 N 150 kHz force impulse in the transmitter area. The frequency of 150 kHz was selected because at such a frequency only the only two modes, symmetrical S\({}_{0}\) and antisymmetrical A\({}_{0}\), may propagate. Second, a 2D method was performed simulating an A\({}_{0}\) and S\({}_{0}\) guided wave modes in excitation zone of 1 mm. This schematic diagram with one excitation zone is presented in Figure 1b. The excitation zone is placed \(x\) = 50 mm, \(y\) = 0 mm from the coordinate origin to at \(x\), \(y\) = 0. In Table 1, some properties corresponding to the computational efficiency of the 3D and 2D methods are compared, which show that the 2D simulation method has a higher speed and needs a relatively smaller memory than the 3D method. Figure 1: A model of the PVC sample used for the simulation: (**a**) for the 3D simulation; (**b**) for the 2D simulation.