## 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
One complete evaluation of error functional \(\Omega\) for a trial solution required on average 14.5 s, a reasonable time considering that each value of \(\Omega\) elaborated by HS-JAYA was obtained by summing over the five values \(\Omega_{1.07\text{KPa}}\), \(\Omega_{1.33\text{KPa}}\), \(\Omega_{1.60\text{KPa}}\), \(\Omega_{1.87\text{KPa}}\) and \(\Omega_{2.13\text{KPa}}\) resulting from structural analyses relative to each pressure level. The construction of the FE model in ANSYS(r) including geometric modeling, meshing, definition of loading/boundary conditions and setting of solution/post-processing options took about 1.5 s of CPU time. Before solving the three variants of the identification problem (35) considered here, the computational efficiency of the HS-JAYA algorithm was preliminarily evaluated by solving the most complicated test case of [27], the mechanical characterization of a bovine pericardium patch (BPP), modeled as a transversely isotropic hyperelastic material. The BPP identification problem included 17 unknowns: 16 hyperelastic constants (a\({}_{1}\), a\({}_{2}\), a\({}_{3}\), b\({}_{1}\), b\({}_{2}\), b\({}_{3}\) for matrix; c\({}_{2}\), c\({}_{3}\), c\({}_{4}\), c\({}_{5}\), c\({}_{6}\), d\({}_{2}\), d\({}_{3}\), d\({}_{4}\), d\({}_{5}\) and d\({}_{6}\) for fibers) and the fiber orientation angle \(\theta\). In order to draw general conclusions, HS-JAYA was compared with many other metaheuristic algorithms including variants of simulated annealing (SA), harmony search (HS) and big bang-big crunch (BBBC), JAYA with approximate line search (relying todifferent extents on error functional gradient information even obtained via finite differences) and elitist strategies. Those variants had in turn outperformed many other metaheuristic algorithms as it is documented in [27]. Standard GA and PSO implementations enhanced by a hill-climbing strategy were also tested in the BPP problem in order to have some comparison with MAs algorithms [57; 58]. On average, HS-JAYA identified material properties of BPP (only 0.25% error on target values) within about only 1100 analyses vs. 1220-1370 analyses required by the other algorithms. GA and PSO enhanced by a hill-climbing strategy did not perform well: error on target material properties still was 0.5% after 1500 analyses. The worst solution of HS-JAYA always was better than the best solutions of all other algorithms in terms of lower errors on target material properties and displacements. Population size selected for HS-JAYA optimizations was only 20 vs. the 30 to 90 population sizes used in [27]. These results confirmed with no shadow of doubt the high efficiency of the proposed HS-JAYA algorithm and justified its use for solving the identification problem (35). All optimizations carried out by HS-JAYA for problem variants 1 and 2 were always completed within (i) 1500 evaluations of the error functional \(\Omega\) for the very large population \(\text{N}_{\text{POP}}=500\), (ii) 700 evaluations of \(\Omega\) for the intermediate size population \(\text{N}_{\text{POP}}=250\), and (iii) 200 evaluations of \(\Omega\) for the very small population \(\text{N}_{\text{POP}}=20\). HS-JAYA always completed the optimization runs entailed by problem variant 3 within 100 evaluations of \(\Omega\). HS-JAYA always found optimum solutions within 20 design iterations for all variants of the identification problem (35) regardless of population size. The present algorithm was also very robust: in fact, the ratio of standard deviation on number of analyses to the average number of analyses required by HS-JAYA in the independent optimization runs never exceeded 2%. Time complexity was not a relevant issue for HS-JAYA; evaluation of error cost functional \(\Omega\) always took almost 80% of total computation time regardless of the number of optimization variables and population size. HS-JAYA outperformed the above-mentioned algorithms of [27] and the GA/PSO algorithms with hill-climbing strategy also in the natural rubber membrane identification problem considered in this study.