## 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
In order to have a rigorous comparison, all optimization runs performed for HS-JAYA were also performed for population-based algorithms (i.e., HS, BBBC, JAYA, hill-climbing-enhanced GA/PSO algorithms) starting from the same initial populations; in the case of SA (a single-point algorithm), three optimizations were performed for each initial population of HS-JAYA, taking as initial points for SA the \(\mathbf{X_{OPT}}\) or \(\mathbf{X_{WORST}}\) or \(\mathbf{X_{aver}}\) designs that, respectively, correspond to the best, worst and average solutions stored in HS-JAYA's population. Whilst HS-JAYA's competitors practically converged to the same material properties and thickness distributions found by the present algorithm (the largest difference from material/thickness parameters identified by HS-JAYA was 3.9%; the largest residual error on experimentally measured displacement components was 6.3%) regardless of NDV and N\({}_{\text{POP}}\), their computational costs were significantly higher than for the present algorithm. In particular, JAYA required on average 15% more analyses than HS-JAYA to complete the optimization process; HS and BBBC required on average between 23% and 26% more analyses than HS-JAYA; and SA required on average 35% more analyses than HS-JAYA. Hill-climbing-enhanced GA/PSO algorithms required on average between 50 and 70% more analyses than HS-JAYA; such a behavior was, however, expected because these algorithms included a much lower amount of elitism than the present algorithm in the transition from exploration to exploitation phases. The 100 analyses' computational cost of HS-JAYA recorded for problem variant 3 was comparable with the 225 to 300 analyses computational costs reported in [66] for identification problems regarding the two-parameter Mooney-Rivlin model and the three-parameter Yeoh model. However, the number of error functional evaluations entailed by the optimization process does not seem to be a critical issue in the hyperelastic materials identification literature, and such information was not very often provided in technical papers [66; 67; 86]. Nevertheless, the selected implementations of some metaheuristic algorithms (e.g., particle swarm optimization used in [67], standard simulated annealing used in [83] and evolution strategies used in [85; 86]) inherently entailed a huge number of finite element analyses (i.e., 10,000 for [67], about 10,000 for [83] and 30,000 to 170,000 for [85; 86]) to solve the identification problem. Table 1 summarizes the main results obtained in the identification problem (35) of the natural rubber membrane. The table presents the average minimum, average maximum and corresponding standard deviations of MR parameters and thickness values identified by HS-JAYA for the three problem variants solved with different population sizes. In problem variant 1, the values of MR constants identified by HS-JAYA ranged from 190 to 215 kPa for a\({}_{10}\) and from 12.5 to 14 kPa for a\({}_{01}\). Hence, the equivalent Young modulus \(\text{E}_{\text{MR}}=4(1+\
u)(\text{a}_{10}+\text{a}_{01})\) of the membrane ranged from 1215 kPa to 1375 kPa with a 13.2% variation, which is fully consistent with the degree of anisotropy of 1.127 found in the preliminary in-plane equi-biaxial test (see Section 4.2). The MR constants identified for problem variants 2 and 3 are very similar (on average, only 0.55% and 0.94% difference on a\({}_{10}\) and a\({}_{01}\), respectively) and practically correspond to the average of maximum and minimum values of hyperelastic constants identified by HS-JAYA for problem variant 1. Figure 13 shows the average distribution (averaged over the 20 independent optimization runs performed for each population size) of the equivalent Young modulus \(\text{E}_{\text{MR}}\) determined by HS-JAYA for the 100 mm diameter natural rubber membrane in problem variant 1. The figure also shows the directions (1) through (6) containing the control paths along which moire data were compared with finite element simulations and specimen thickness was measured.