## 4 Results and Discussion
### Detailed Mechanical Characterization of the 100 mm Diameter Membrane
#### 4.3.2 Solution of the Inverse Problem: FE Analysis and Metaheuristic Optimization
Interestingly, the HS-JAYA results indicate that the equivalent Young modulus is not constant along any of the control directions and, above all, values of Young modus of each couple of orthogonal directions (i.e., (1)-(4), (2)-(5) and (3)-(6)) are in a different ratio. For example, it holds that \(\text{E}_{\text{MR}}^{(1-)}/\text{E}_{\text{MR}}^{(4+)}=1.2\), \(\text{E}_{\text{MR}}^{(2-)}/\text{E}_{\text{MR}}^{(5-)}=1.05\) and \(\text{E}_{\text{MR}}^{(3+)}/\text{E}_{\text{MR}}^{(6-)}=0.92\) where the \"\(-\)\" sign denotes the direction part with at least one negative coordinate x or y. This confirms the non-isotropic mechanical response of the tested specimen. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Identified Properties** & **Problem Variant 1** & **Problem Variant 2** & **Problem Variant 3** \\ \hline \(N_{\text{TOP}}=20\) & & & \\ \(a_{10}\text{min}\) 191.638 \(\pm\) 1.642 & \(\dfrac{N_{\text{TOP}}=20}{204.852\pm 2.089}\) & \(\dfrac{N_{\text{TOP}}=5}{203.763\pm 2.011}\) \\ \(a_{01}\text{min}\) 12.568 \(\pm\) 0.124 & \(a_{01}\) 13.412 \(\pm\) 0.137 & \(a_{01}\) 13.279 \(\pm\) 0.126 \\ \(a_{01}\text{max}\) 13.834 \(\pm\) 0.147 & & \\ \hline \(N_{\text{TOP}}=250\) & & & \\ \(a_{10}\text{min}\) 191.727 \(\pm\) 1.526 & \(\dfrac{N_{\text{TOP}}=250}{204.819\pm 2.067}\) & \(\dfrac{N_{\text{TOP}}=10}{203.692\pm 1.989}\) \\ \(a_{01}\text{min}\) 12.675 \(\pm\) 0.113 & \(a_{01}\) 13.356 \(\pm\) 0.128 & \(a_{01}\) 13.234 \(\pm\) 0.117 \\ \(a_{01}\text{max}\) 13.743 \(\pm\) 0.145 & & \\ \hline \(N_{\text{TOP}}=500\) & & & \\ \(a_{10}\text{min}\) 191.563 \(\pm\) 1.490 & \(\dfrac{N_{\text{TOP}}=500}{204.833\pm 1.996}\) & \(\dfrac{N_{\text{TOP}}=20}{204.893\pm 1.996}\) & \(\dfrac{N_{\text{TOP}}=20}{203.728\pm 2.004}\) \\ \(a_{01}\text{min}\) 12.497 \(\pm  
Thickness distributions identified by HS-JAYA for variants 1 and 2 of the inverse problem (35) including the 108 local thickness values as optimization variables also were consistent with experimental measurements. Figure 14a shows the measured thickness distribution for the 100 mm diameter membrane. The measured maximum and minimum thickness values were, respectively, 1.073 mm and 0.930 m, while the standard deviation of the measured values with respect to the 1 mm nominal thickness was 0.0051 mm (i.e., about 0.55% of minimum thickness measured experimentally). Thickness distribution is consistent with observed displacements (Figures 8-10): the membrane is stiffer in the x\(<\)0 and y\(<\)0 regions, which leads to larger deformations in the rest of the specimen. Further evidence of this can be gathered from the distribution of equivalent Young modulus E\({}_{\text{MR}}\) of Figure 13: elastic parameters are larger especially in the (x,y) \(<\) 0 region. Figures 14b and 14c, respectively, show the percent errors on membrane thickness distributions identified by HS-JAYA in variants 1 and 2 of the inverse problem. For the \(t\)th generic membrane sector (see the schematic of Figure 11), percent error on identified thickness is defined as \((\text{t}_{\text{s}}\text{identified}-\text{t}_{\text{s}}\text{measured})/ \text{t}_{\text{s}}\text{measured}\). The largest magnitude errors (either positive or negative) are plotted in the figure. It can be seen from Figure 14 that errors on identified local thicknesses practically coincided for the two problem variants and always ranged between \(\pm\)0.3%, almost three orders of magnitude less than measured values. Interestingly, the values of MR constants determined by HS-JAYA for problem variant 2 practically correspond to the average of the MR parameter distributions determined for problem variant 1. Hence, the HS-JAYA algorithm correctly identified the thickness variations over the whole membrane. The statement made above was confirmed by the statistical analysis of HS-JAYA's optimized solutions.