## 2 Numerical Investigation  
Propagation of the guided subsonic A\({}_{0}\) mode in the mica paper sample was simulated by the Abaqus/CAE 2016.HF2 software package (Dassault Systemes, Johnston, Rhode Island, United States). Dimensions of the mica sample were \(160\times 160\) mm. For reduction of the computational time, only a quarter of the paper with symmetry boundary conditions was modelled. The modelled structure is presented in Figure 1.  
During the simulation it was assumed that there are zones with different elastic properties and thickness. The real MICA paper samples contain mica ore muscovite in optically lighter zones and phlogopite in darker zones, the elastic properties and thickness of which are different. In the darker zone thickness of the sample is \(d=35\)\(\mu\)m, the [PERSON]'s modulus \(E_{\text{D}}=16\) GPa, the [PERSON]'s coefficient \(\
u=0.25\), \(\rho=771\) kg/m\({}^{3}\).  
The simulation step in the time domain was \(dt=1\)\(\mu\)s. The excitation was applied at a rectangular zone \(5\times 1\) mm\({}^{2}\), the centre of which was at \(x=0\), \(y=0\). The excitation was introduced as the uniformly distributed 1 N 44 kHz force impulse in the transmitter area. The frequency of 44 kHz was selected because at such a frequency only the slow A\({}_{0}\) mode may propagate. The waveform and the spectrum of the excitation force are shown in Figure 2a,b.  
Figure 1: A model of the mica sample used for the simulation.  
The model of the mica paper was meshed using shell elements. The shell thicknesses are 30 \(\upmu\)m in the lighter zones and 35 \(\upmu\)m in the darker zones. The dimension of the finite elements was 0.1 mm. Such a dimension is close to 1/20 th of the wavelength \(\lambda\) = 2 mm at the analysed frequency.  
In order to demonstrate efficiency of the proposed reconstruction methods of spatially non-uniform elastic properties three different cases with a hypothetical simple geometry were simulated--the mica paper sample with two zones of rectangular shape with different thickness and elastic properties (Figure 3), the sample with a single circular non-uniformity, and the sample with multiple circular non-uniformities of different diameters.  
Figure 2: The waveform (**a**) and the spectrum of the excitation force (**b**); n.u. represents normalized units.  
The simulated spatial distributions of the off-plane l particle velocity of the mica paper with the non-uniformity shown in Figure 3 at two different time instants \(t=150\)\(\upmu\)s and \(246\)\(\upmu\)s after excitation are shown in Figure 4.  
As can be seen the waves are more strongly radiated along \(x\) axis direction due to directivity properties of the excitation zone. The weak pattern of the wave reflected by the boundary between zones with different properties can be observed at the time instant \(246\)\(\upmu\)s also. The waveforms of normal displacements at two different distances from the excitation area are shown in Figure 5 in which the tail of the main pulse of the signal is reflected by the border between the zones with different properties can be seen (Figure 5a).  
Figure 4: Spatial distributions of the normal particle velocity at two different time instants: (**a**) 150 \(\upmu\)s; (**b**) 246 \(\upmu\)s.  
Figure 3: Sample of the mica paper with two zones with different elastic properties.