## 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
The error functional \(\Omega\) thus defined with Equation (35) was well posed, avoiding bias phenomena that may propagate errors in the minimization process of error functional even when target quantities are derived from synthetic patterns (see, for example, the very recent study [65] that analyzed performance of a FEMU approach operating on DIC strains). Three population sizes, comprising, respectively, 20, 250 and 500 candidate designs, were selected for performing metaheuristic optimizations relative to the first two variants of the inverse problem described above: hence, the N\({}_{\text{POP}}\)/NDV ratio ranged from 0.0617 to 1.543 for the first problem variant with 324 design variables, and from 0.182 to 4.545 for the second problem variant with 110 design variables. For the third problem variant including only two design variables, population size was 5, 10 or 20: the N\({}_{\text{POP}}\)/NDV ratio hence ranged from 2.5 to 10. This was conducted to verify the insensitivity of HS-JAYA convergence behavior to the size and composition of the initial population and, in summary, to confirm the parameter-free nature of the metaheuristic optimizer selected for this study. The HS-JAYA optimization algorithm for solving the inverse problem (35) was implemented in the ANSYS(r) parametric language. The main code defined the problem constraints (including design variables bounds) and perturbed material/stiffness properties of the membrane following the steps of HS-JAYA formulation. Each time a new trial combination of hyperelastic constants and thicknesses was defined, the main code prompted ANSYS(r) to create the new FE model of the membrane and perform structural analysis. In post-processing, computed displacements were compared with target values to evaluate error functional \(\Omega\). The new trial design was accepted/updated/rejected based on its quality as described in Section 3. The optimization process was terminated when the error functional and design vector varied by less than 0.0001 in two subsequent iterations. In order to account for the metaheuristic nature of HS-JAYA, each optimization run was repeated 20 times starting from various randomly generated populations. Equation (14) was used to generate initial populations covering the whole design space. However, the following cases were also considered in order to force HS-JAYA to start optimizations very far from target solution: (i) MR parameters and membrane thicknesses close to their upper bounds; (ii) MR parameters and membrane thicknesses close to their lower bounds; (iii) MR parameters close to their lower bounds and membrane thicknesses close to their upper bounds; and (iv) MR parameters close to their upper bounds and membrane thicknesses  
Figure 12: FE model of the 100 mm diameter natural rubber membrane: (**a**) mesh; and (**b**) loads and kinematic constraints. The six control paths (corresponding to in-plane loading directions) including thickness measurement locations also are sketched in the figure. close to their lower bounds. The final solution given in output by HS-JAYA was averaged over those obtained in the independent runs. Interestingly, standard deviations of identified material properties and stiffness distribution over the different optimization runs always were less than 1.1% for all problem variants regardless of population size, thus confirming the robustness of the HS-JAYA algorithm selected in this study for solving the inverse problem (35). All computations were performed on a Toshiba Satellite L50-B-24C portable computer with an Intel(r) CoreTM i7-5500 CPU and 8 GB of RAM memory in the Windows 10 environment. Evaluations of error functional for the inverse problem solution counted for more than 95% of the total computation time because of the nonlinearity of the structural analyses.