# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 4 Validation Using the International Parallel Test Benchmarks
### Shear Modulus Extraction
#### 4.2.3 Automatic Shear Modulus Extraction Using GeoHyTE  
In GeoHyTE, the definition of the computational model follows the steps described in Section 3.2. The plots given there as GUI examples are obtained on this test case and are not reproduced here to avoid redundancy. The output signal, presented in Figure 13, is loaded into the first GUI (Figure 3), and the tentative arrival time, \(t_{art}=0.920\) ms, and correlation window of 1.220 ms are selected. The arrival time corresponds to an initial estimate of the shear modulus of \(G_{0}^{1}=61.68\) MPa. The correct choice of the tentative arrival time is not of great importance, as discussed at length in Section 5.1.  
The finite element model is defined in the following user interfaces, as presented in Figures 4-9. The subsequent run of the model converges in two steps, producing the output presented in Figures 17 and 18.  
After the first run, the instantaneous correlation between the numerical and experimental output signals is scored at \(-0.336\), as indicated by a blue circular marker in the plot on the right, which is a low correlation. However, the maximum correlation, found \(7.085\times 10^{-2}\) ms to the right, looks more promising, with an estimate value of 0.621. The tentative arrival time is incremented with the time lag to the maximum correlation, that is, \(t_{art}=0.920+7.085\times 10^{-2}=0.991\) ms, to which a new tentative shear modulus of \(G_{0}^{2}=53.20\) MPa corresponds. This is the value used in the second iteration. These findings are automatically logged in the lower part of the output zone.  
After the second run (Figure 18), the instantaneous correlation is also the maximum correlation, with a score of 0.731, meaning that the fixed point of the shear modulus is found. Its final value is thus \(G_{0}=53.20\) MPa, which is delivered to the user.  
The time-history of the horizontal displacements at the tip of the receiver, normalised to their maximum amplitude, is presented in Figure 19. For a better illustration of the finite  
Figure 16: Unwrapped phase function for the reference run.  
Figure 15: Phase function for the reference run.  
element solution, the horizontal displacement solutions over the whole domain are plotted in Figure 20 at key instants \(t_{1}\) to \(t_{6}\), indicated in Figure 19.  
Figure 17: Results after the first iteration.  
Figure 18: Results after the second iteration.  
Figure 19: Time–history of normalised displacements at the tip of the receiver.  
Figure 20: Horizontal displacement fields obtained with the reference test model.  
The first two instants correspond to the generation of the shear wave near the transmitter and to its early propagation after the motion of the transmitter ceases. The shear waves are clearly visible and their propagation from the transmitter to the receiver can be easily followed visually throughout the plots. The first significant oscillation in the numerical output occurs around points \(t_{3}\) and \(t_{4}\) and corresponds to the early arrival of compression waves reflected from the lateral envelope of the sample (Figure 20c,d). Although compression waves trigger lateral motion of smaller amplitude than shear waves, their criss-cross propagation through the sample can also be followed visually. Finally, the arrival of the shear wave and its reflection from the top platen are recorded in \(t_{5}\) and \(t_{6}\), respectively, as apparent from Figure 20e,f. The quality of the finite element solution can be evaluated by the spurious discontinuities of the displacement field between the finite elements, as hybrid-Trefftz finite elements are neither locally continuous nor locally equilibrated. However, no visible discontinuities are noticeable in any of the plots in Figure 20, meaning that the solution converges. It is also noted that the applied null lateral displacements on the lateral walls are correctly recovered by the model in all steps, which is another mark of convergence.