# Multi-Objective Pareto Optimization of Tensile Membrane Architecture for Energy Harvesting
## 4 Discussion  
Figures 4 and 5 show the results of 18 multi-objective Pareto optimizations. Five optimization simulations were conducted, one at a time, on a total of two computers over 2 d, taking a total of 4 d. The x-axis of the graph indicates the stiffness, and the y-axis indicates the acceleration. To unify the graph range between structures of the same type and improve the legibility of the graph, we removed outliers whose acceleration values were too low. The number of alternatives removed is shown at the bottom left of each graph. The polyline in each graph represents the 30 th-generation Pareto-front solutions. The three highlighted points in the graphs represent two extremes and middle design alternatives, which were used to assess the optimization, and the 3D models of the three alternatives are presented at the bottom of each graph. As provided the most energy generation among the alternatives, and the _Cs_ provided the highest stability (Figures 4 and 5). One that has its acceleration value closest to the average acceleration value of As and Cs was selected as B. For each representative alternative, detailed values are presented on the right side of the model. In all the graphs, an inverse relationship between the stiffness and acceleration is clearly observed. Similar to the previous study [12], the stability and energy efficiency are inversely correlated, regardless of the wind environment. The optimal graph for the hypar shape was convex, and the rest of the graphs were straight-like concave. Thus, the graphs for the conic and barrel-vault shapes differ substantially from that for the hypar shape. Because of the convex shape, the hypar is significantly affected by the material stiffness on small stiffness values, and the amount of electricity generated decreases as the material becomes stiffer. In comparison, the conic and barrel-vault shapes are constantly affected by material stiffness, and even if the material is stiff, it is not significantly affected by the amount of electricity generated. The selected alternatives of Figures 4 and 5 exhibit efficient geometric shapes that are hardly affected by the wind pattern. Among a total of 54 alternatives, regardless of location in Inner Mongolia or USA, the shape variable that ensures high energy efficiency appears. Thus, the selected alternatives have common shapes. Dominating shapes that showed commonalities on most optimized alternatives are the hypar with former and later heights of 7 m; the conic with ring radius of 6 m and ring height of 7 m; and the barrel vault with former and later y-radiuses of 7 m. The selected alternatives with dominating shape were marked with red dots in Figures 4 and 5. On the other hand, only the hypar selected alternatives in Texas have an optimized shape different from the dominating shape. The non-dominant hypar comprises the former height of 7 m and the later height of 0 m or the opposite. It shows that the optimized shape may be different depending on the wind environment. The shape of selected alternatives in Texas or the dominating shapes have similarities in which the height/side length ratio of geometry is large. Since the wind blows horizontally in the simulation environment, alternatives of shapes with a large height/side length ratio appear to be dominating heavily. In conclusion, the dominating shapes in the five environments, except the USA hypar, have the highest energy harvesting in any stiffness or wind environment. \begin{table}
\begin{tabular}{c c} \hline
**Optimizing Engine** & **Wallacei X** \\ \hline Generation size & 25 \\ \hline Generation count & 30 \\ \hline Crossover probability & 0.9 \\ \hline Crossover distribution index & 20 \\ \hline Mutation distribution index & 20 \\ \hline Number of genes & 3 \\ \hline Number of values & 5752 (conic) and 5837 (hypar and barrel vault) \\ \hline Number of fitness objectives & 2 \\ \hline \end{tabular}
\end{table} 
Table 1: Optimizing parameters. Figure 4: Pareto-optimization results for Inner Mongolia, China.