## 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
Since 20 optimizations were performed for all problem variants starting from different initial populations, pairs of thickness distributions (each including the 108 thickness values identified by HS-JAYA) corresponding to the \(p\)th (\(p=1,\dots,20\)) optimization run carried out for variants 1 and 2 were compared in terms of mean and standard deviation. Interestingly, these two parameters always matched, indicating that all solutions obtained for variants 1 and 2 belong to the same population. The same result was obtained for all population sizes N\({}_{\text{POP}}\) = 20, 250 or 500 worked out by HS-JAYA. Another statistical analysis investigated if any difference between thickness distributions could have been derived from the complexity of design space of problem variants 1 and 2 that, respectively, included 324 and 110 design variables. However, this scenario was excluded by running three Wilcoxon tests that found a very high correlation between all solutions within a level of confidence of 0.05. The Wilcoxon test gives a \(p\)-value stating whether a statistical difference is significant or not; the smaller the \(p\)-value is, the greater the difference between solutions will be. Here, the first Wilcoxon test compared two populations of 20 elements including the maximum (i.e., largest positive) errors on thickness recorded for problem variants 1 and 2. The second Wilcoxon test compared the two populations formed by minimum (i.e., largest negative) errors on thickness. The third Wilcoxon test compared two populations including median error values on thickness for problem variants 1 and 2. Remarkably, \(p\)-values always were much higher than 0.05, thus confirming that there is no statistical difference between solutions of problem variants 1 and 2. The same conclusion was reached for all population sizes N\({}_{\text{POP}}\) = 20, 250 or 500 worked out by HS-JAYA. Figure 15 shows the distribution of total displacements u\({}_{\text{tot}}\) computed by ANSYS\({}^{\text{\text{\textregistered}}}\): the plotted map is the average of the u\({}_{\text{tot}}\)-displacement maps relative to the optimization runs performed for each combination of problem variant and population size. As expected, the distribution shown in Figure 15 is very similar to the w-displacement (the dominating displacement) map measured experimentally (see Figure 8c). The point of maximum deformation does not coincide with the center of the membrane and lies in the x,y \(>\) 0 quadrant. Contours are not perfect circles but ellipses with the axes slightly rotated with respect to the X-Y reference system. The average magnitude of the largest total displacement computed by ANSYS\({}^{\text{\text{\textregistered}}}\) was 13.969 mm (see Figure 15), only 0.12% different from the experimentally measured value of 13.952 mm. Figure 16 compares in detail displacements \(\upmu\), \(\
u\) and \(w\) measured by the one-shot projection moire setup described in this study with those computed by ANSYS(r) in correspondence with the average optimized solutions for the three variants of the identification  
Figure 14: (**a**) Thickness distribution measured for the 100 mm diameter membrane; (**b**) distribution of largest magnitude % error on identified thicknesses in problem variant 1 with respect to measured thickness values; and (**c**) distribution of largest magnitude % error on identified thicknesses in problem variant 2 with respect to measured thickness values. Figure 15: Distribution of total displacements (\(\upmu_{\text{tot}}\)) computed by ANSYS®: average for the different solutions of problem variants 1, 2 and 3.  
problem (35). In particular, Figure 16a refers to the u-displacements of the horizontal diameter of the membrane (i.e., control path 1 lying on the X-axis), while Figure 16b refers to the v-displacements of the vertical diameter of membrane (i.e., control path 4 lying on the Y-axis); furthermore, Figure 16c,d refer to w-displacements of control paths 1 and 4, respectively.