## 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
Experimental data are represented by a double-sided line embedding the nine in-plane displacement distributions corresponding to the three problem variants solved with three different population sizes. Similar results were obtained for the other control paths sketched in Figure 12b and are not shown in this paper in order to save space. The displacement distributions computed by ANSYS(r) present some oscillations as they were averaged over the 20 independent optimization runs performed for each combination of problem variant and population size. However, all trends of numerically reconstructed displacements practically show the same features. The highest residual errors on displacements were recorded for problem variant 3, which included only 2 MR constants as design variables: (i) for u-displacements, 1.883 \(\div\) 2% vs. only 1% of problem variant 1; (ii) for v-displacements, 1.806 \(\div\) 2.096% vs. only 1.060 \(\div\) 1.170% of problem variant 1; and (iii) for w-displacements, 2.436 \(\div\) 3.333% vs. only 1.25 \(\div\) 2% of problem variant 1. Overall, the largest error on total displacement never exceeded 2.5% for problem variant 1, 3.7% for problem variant 2 and 5.6% for problem variant 3. Maximum errors were distributed in a rather uniform way in the case of in-plane displacements. However, for out-of-plane displacements, maximum errors were localized either near the constrained membrane boundary or near the points where in-plane displacements reached their highest magnitude. This confirms the need for precisely matching the deformed position (x\({}^{\prime}\),y\({}^{\prime}\)) of each pixel of the moire pattern image with the out-of-plane displacement component w.  
Interestingly, the maximum residual error on displacement components never went below 1% for all variants of the identification problem (35). In order to prove that this was due not to some weakness in the formulation of the HS-JAYA algorithm, the identification problem (35) was solved in silico by computing \"synthetic\" displacement fields for the average solution of each problem variant. Hence, the optimizer had to recover the numerically generated target displacement fields. Remarkably, HS-JAYA always converged to the target material/structural properties reproducing displacement fields within at most 0.023% residual error. The observed residual errors in the hybrid numerical-experimental identification process hence were fully consistent with the level of uncertainty normally entailed by FEMU-based characterization. Optically measured displacements represent a good target because PM accuracy may reach a very small fraction of the sensitivity of the experimental setup. Hence, results of the identification process documented in this study may be judged satisfactory. Figure 16: _Cont._  
Figure 16: Experimentally measured displacements vs. those computed by ANSYS(r) for the identified material/structural properties in the different variants of problem (35) solved in this study: (**a**) u-displacement (horizontal diameter, control path 1); (**b**) v-displacement (vertical diameter, control path 4); (**c**) w-displacement (horizontal diameter, control path 1); and (**d**) w-displacement (vertical diameter, control path 4).