# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 5 Robustness of GeoHyTE Predictions
### Robustness to Imperfect Finite Element Solutions  
The problem analysed in Section 4.2.3 is solved here using inadequate mesh and basis refinements to assess the consequence of erroneous finite element solutions on the output of the iterative model updating process. The algorithmic parameters chosen for the reference setup described in Section 4.2.3 are adopted to endorse direct comparison. The first tentative arrival time is \(t_{arr}=0.920\) ms and the correlation window is taken until after the largest oscillation in the output signal, at \(t_{corr}=1.220\) ms.  
The finite element model is constructed using the purely rectangular geometry presented in Figure 27, meaning that the transmitter border element is not explicitly included in the model. The domain is discretised into 40 rectangular elements of equal dimensions, with no additional refinement in the zone with large gradients of the solution. The elements are 20 mm by 25 mm in size. Very coarse approximation bases are adopted in the finite elements and on their essential boundaries. For the former, Bessel bases of the third order are adopted, while for the latter, linear shape functions are used. The total number of degrees of freedom of the model is 1348. For comparison, it is noted that the reference model presented in Section 4.2.3 features 198 finite elements and 14296 degrees of freedom. The computational time for the reference model is 10-12 times higher than for this model.  
Because the transmitter border element is not included in the model, the shear wave is triggered by the lateral (tangential) displacement of the boundary located in the middle  
Figure 27: Simplistic geometry and coarse mesh used to obtain an imperfect finite element solution (dimensions in millimetres).  
Figure 26: Update function \(\mathcal{G}(G_{0})\) and representation of the convergence process for starting point \(A\).  
of the bottom side of the domain, as indicated in Figure 27. The lateral displacement has a parabolic variation in the horizontal direction, null on the extremities of the boundary and equal to \(10^{-6}\) m in the middle. All other exterior boundaries are unconfined in the tangential direction and fixed in the normal direction.  
As expected, the finite solution obtained in this way is of poor quality. The horizontal displacement time-history at the tip of the receiver is presented with a solid line in Figure 28, overlapping the solution obtained with the fine refinement (dotted line). The displacement values are normalised to the maximum absolute displacement of the refined solution.  
In Figure 29, the horizontal displacement solutions over the whole domain are plotted at instants \(t_{1}\) to \(t_{6}\), identified in Figure 28. Note that these are the same instants used to plot the solutions for the refined model in Figure 20.  
It is clear that the finite element solution obtained with the coarse model is far worse than that obtained with the finer refinement, especially in the bottom region, where the solution gradients are large. Displacement fields are markedly discontinuous between adjacent elements in Figure 29a,b (the mesh was overlapped to stress the internal boundaries), and spurious solution oscillations are also identifiable in the other plots. However, they tend to diminish over time as the gradients of the fields decrease, meaning that the output signal is only slightly affected by these issues (see Figure 29d,f). This is a consequence of the fact that the problem is solved in a single time step [14], meaning that the discontinuity of the solution at a certain time is not transmitted to the next time step as an initial condition. The iterative model updating process converges in only two iterations to shear modulus estimate \(G_{0}=52.30\) MPa, consistent with the value obtained for the reference case, \(G_{0}=53.20\) MPa.  
Figure 28: Time–histories of horizontal displacements at the tip of the receiver for the coarse refinement (solid line) and fine refinement (dotted line).